Analytic Geometry MAT 1035
|
|
- Owen Harper
- 5 years ago
- Views:
Transcription
1 Analytic Geometry MAT WEEKLY PROGRAM - The first week of the semester, we will introduce the course and given a brief outline. We continue with vectors in R n and some operations including addition of two vectors. The second week, we talk about inner product, vector product (cross product) and their geometric interpretations. We also give the projection of a vector along another one and perpendicular and parallel vectors. Furthermore we give some related theorems and some (un)solved examples. Furthermore, we are going to give the mixed product with their geometric interpretation. If time permits we will start the new chapter LINE. In the tutorial section, you will see points of division, and solve the following exercises. Tutorial. Let V be a vector space and <, > an inner product on V. a) Show that < 0, u >= 0 for all u in V. b) Show that < u, v >= 0 for all v in V, then u = 0.. Let <, > be standard inner product on R. [ ] [ ] a) Let α = and β =. Find a vector γ such that < α, γ >= and < β, γ >= 3. b) Show that α =< α, e > e + < α, e > e for any α in R. (Here, the vectors e and e are standard basis vectors of R.) [ ] 3 3. Find the norm of α = with respect to 4 a) the usual inner product. b) the inner product given as < x, y >= x y x y x y + 3x y. 4. Sketch a directed line segment in R 3 representing each of the following vectors:
2 a) A vector with tail (beginning point) A(,, 5) and head (end point) B(3,, ). b) A vector with tail (beginning point) A(,, 4) and head (end point) B(, 4, 3). 5. Find the vector AB, and their lengths, and also sketch them in R 3 if a) A(,, 0) and B(,, 7). b) A(5,, 3) and B(,, 0). 6. Determine the beginning point of the vector (,, 3). 7. For what values of a, b and c are the vectors equal? 6 0 a b a b 6 whose end point is and 8. Compute u + v, u v, u 3 v, 3 u + v and sketch them if a) u = and v = 4 3. b) u = 0 and v = 4. a + b c 9. Show that AB + BC + CA = 0 for a triangle ABC. 0. Show that AD = ( AC + AB) for the median vector AD of a triangle ABC.. In the figure, the points A, B and C are collinear. If AC = 3 AB and DC = k AD + pdb then k p =? A B C D
3 . Let G be a barycenter of a triangle ABC. Show that a) GA + GB + b) for any point K, GC = 0 KA + KB + KC = 3 KG. 3. Let ABC be a triangle, and D and E be the midpoints of AB and AC. Then show that DE is parallel to BC ( DE BC) and DE =. BC 3 4. If the vectors u = m and v = 3m are parallel, then find the n m 8m values of m and n. [ ] 5. Let u = and v = v ω = 3. [ ] be given. Find ω such that u ω = and 6. Find the norm of given vectors, and also find the unit vectors in directions of them. 3 a) u = 3 b) v = Find the projection of the vector u = 8. Find the angle between the vectors u and v if a) u =, v =. b) u = 0, v = 0. onto v = 9. In R n if u + v = u v then find the angle between the vectors u and v
4 0. Let a, b and c be nonzero vectors. If a b = a c, then can we necessarily say that we have b = c?. Prove the following inequalities: a) Triangle inequality : a + b a + b for all a, b and c in R n. b) Schwartz inequality : a b a b for all a, b and c in R n.. Let a and b mutually perpendicular non-zero vectors. Show that for any number λ, a + λ b a. 3. For any vectors a, b in n-space, prove the followings: a) a + b + a b = ( a + ) b. b) a + b a b = 4 a b. 4. If possible, find a, b and c so that v = and y =. a b c is orthogonal to both x = 5. Let ABC be a triangle. Prove the cosine theorem, i.e. BC = AC + AB AC AB cosâ AC = BC + AB BC AB cos ˆB 6. Compute the followings: AB = BC + AC BC AC cos Ĉ. a) e e b) e e 3 c) e e d) e e 3
5 e) e 3 e f) ( e e ) e 3 g) ( e e ) e 3 h) x y if x = i) u v if u = 5 8 and y = and v = 7. If u v = u ω then can we necessarily say that v = ω? 8. Given u =, v = 3 6 and ω = compute a) ( u v) ω b) u ( v ω) c) ( u v) ω d) u ( v ω) 9. Let u, v and ω be arbitrary vectors in 3-space. Is there any difference between ( u v) ω and u ( v ω)? 30. Find a nonzero vector which is perpendicular to both u and v given as follows: a) u = and v =. 3 0 b) u = c) u = and v = and v = Let u, v and ω be mutually perpendicular vectors in R 3. Show that ( u v) ω = 0.
6 3. Let u, v, ω and t be arbitrary vectors in 3-space. Prove the following expansions: a) ( u v) ( ω t) = ( u, v, t) ω ( u, v, ω) t = ( u, ω, t) v ( v, ω, t) u b) ( u v) ( ω t) = ( u ω)( v t) ( v ω)( u t) 33. Find the coordinates of midpoint of the line segment joining the points P (3, 7) and P (, 3). 34. Find the coordinates of the point P which divides the line segment joining the points P (, 5) and P (4, ), in the ratio at 6 and, respectively Find the slope and direction cosines of the line joining the points P (, ) and P ( 5, 3). 36. Find the least angle of the triangle ABC, the coordinates of whose vertices are A(, 4), B( 5, ) and C(0, 6). 37. Find the angle between the directed lines joining P (, 3), P ( 4, 3), and P 3 (, 0) and P 4 ( 5, 6) by slopes and direction cosines. 38. Show that the two lines joining P (7, 5), P (, ), and P 3 (4, 3) and P 4 (, 0) are perpendicular. Bedia Akyar Moller
7 Analytic Geometry MAT WEEKLY PROGRAM 3-4 I am going to remind you scalar product, vector product and mixed product and their geometric interpretations. Furthermore, I will give some related theorems and solve some examples. I am going to start the new chapter line. I will start with direction numbers, direction angles (direction cosines), perpendicular or parallel lines. Then I will give the equation of a straight line, the intercept form of a straight line and the normal form of the equation of a straight line in plane and the angle between two lines in terms of tangents and their direction cosines. Solve the following exercises: Tutorial. Find the slope and direction cosines of the line which is perpendicular to the line joining the points P (, 4) and P (, ).. Find the angle between the directed lines joining P (, 3), P ( 4, 3), and P 3 (, 0) and P 4 ( 5, 6) by slopes and direction cosines. 3. Given the triangle whose vertices are A( 5, 6), B(, 4) and C(3, ) derive the equations of the three medians and solve algebraically for their point of intersection. 4. Find the intercepts of the line perpendicular to x + 3y 7 = 0 and passing through the point (, 6). 5. Find the equations of the lines through (, 6) if the product of the intercepts for each line is. 6. What are the direction cosines of a line perpendicular to x 5y + 3 = Find the equations of all the lines with slope m = 3 4 will occur an area 4 unit with the coordinate axes. such that each line 8. Determine the parameter k such that a) the line 3kx + 5y + k = 0 passes through A(, 4).
8 b) the line 4x ky 7 has the slope 3. c) the line whose equation x + ky + 3 = 0 shall make an angle 45 o with the line x + 5y 7 = Find the perpendicular distances from origin to the lines a) x 3y + 6 = 0 b) 5x 8y 5=0 0. Find the equation of the line through the point of intersection of the lines x 3y + = 0 and x + 5y 9 = 0 and whose distance from the origin is 5.. Calculate the distance d from the line 5x y 3 = 0 to point (, 3). Are the point (, 3) and the origin on the same side?. Determine the equations of the bisectors of the angles between the lines l : 3x + 4y = 0 and l : x 5y + 5 = Write the equations of the line whose parametric equations are x = 5 t, y = 3 + t, z = 3t in symmetric form. 4. Write the equations of the line through the points (7,, ) and (3,, 4). 5. Write the equations of the line parallel to x = y + 4 = z 3 (4,, 3). and through 6. Discuss the intersection of the lines l and l if a) l : x + = y = z 3 and l : x + = 6 y = z +. b) l : x + = y = z 3 and l : x 4 = y = z 5. c) l : x + = y = z 3 and l : x 3 = y 4 = z. Bedia Akyar Moller
9 Analytic Geometry MAT WEEKLY PROGRAM 5-6 We have started the chapter straight line and seen direction numbers, direction angles (direction cosines), perpendicular or parallel lines. We have also given the symmetric and parametric equations of a straight line, the slope of a line, the intercept form of a line, the normal form of the equation of a straight line in R, the angle between two lines, division of points. We am planning to start the new chapter plane. We will give the vector form of the equation of a plane, the angle between two planes and we will also give other forms of the equation of a plane. I will also give the distance from a point to a plane. We will mention some situations between planes and lines we will again talk about the intersections of two planes and some examples by different methods. We will see intersections of three planes and intersections of lines and planes and finally the specialized distance formulae and finish the chapter plane. If time permits, we will start the new chapter Locus Problems. We will also give circles determined various conditions. We will do some examples. I highly recommend you to read Matematik Dunyasi, 005 Yaz, Guz, Konikler. Solve the following exercises:. Show that the two lines passing through P (7, 5), P (, ) and P 3 (4, 3), P 4 (, 0) are perpendicular.. Find the area of the triangle A(, ), B(5, 3) and C( 8, 0). 3. Show that three points A(, 5), B(6, ) and C( 4, ) are colinear (that is, they lie on a line). 4. Prove that the diagonals of a rectangle are equal. 5. Prove that the diagonals of a parallelogram bisect each other. 6. Prove that the medians of a triangle intersect in a point 3 along one from a vertex toward the opposite side. of the distance 7. Find the intercepts of the line perpendicular to x + 3y 7 = 0 passing through the point (,6). 8. What are the direction cosines of a line perpendicular to x 5y + 3 = 0.
10 9. Find the equations of all lines with slope m = 3 such that each line will 4 occur an area 4 unit with the coordinate axes. 0. Determine the parameter k such that i) The line 3kx + 5y + k = 0 passes through A(, 4). ii) The line 4x ky 7 = 0 has the slope 3.. Write the equations of the line x = 5 t, y = 3 + t, z = 3t in symmetric form.. Write the equations of the line through (7,, ) and (3,, 4). 3. Write the equations of the line parallel to x (4,, 3). = y+4 = z 3 and through 4. Write the equations of the line through (,, 8) and perpendicular to the plane 3x + 7y z + = Write the equations of the line through (3, 4, 0) and perpendicular to the xy-plane. 6. Find the equation of the plane passing through the three points (,, ), (,, ), (, 0, ). 7. Find the equation of the plane perpendicular to the line joining (,, 0) and (3,, 5) and passing through (, 5, 8). 8. Reduce 3x + y z + 5 = 0 to normal form. 9. Reduce x 5y + z 3 = 0 to intercept form and write the coordinates of the intercepts. 0. Write equation of the plane passing through (,, ) and (3, 5, 4) which is perpendicular to the xy-plane.. Find the equation of the plane perpendicular to the plane x y + z 5 = 0, parallel to the line whose direction cosines are,, 5 and passing through (, 4, ).. Find the equation of the plane parallel to the 3x y + 6z + 5 = 0 and passing through (, 4, ). 3. Find the equation of the plane parallel to the xz-plane through (, 4, 6). 4. Find the equation of the plane parallel to the xy-plane and units from it.
11 5. Find the equation of the plane with intercepts,, Find the distance from the plane x + y z + = 0 to the point (0,, 3). 7. Determine the angle between x + y z + 7 = 0 and 3x y 6 = 0. Bedia Akyar Moller
12 Analytic Geometry MAT WEEKLY PROGRAM 7-8 We have started the chapter plane and given the vector form of the equation of a plane, the angle between two planes and we have also given other forms of the equation of a plane. I have given the distance from a point to a plane. We have mentioned some situations between planes and lines. We have talked about the intersections of two planes and some examples by different methods. We HAVE NOT SEEN intersections of three planes and intersections of lines and planes and the specialized distance formulae and finish the chapter plane. YOU ARE RESPONSIBLE OF THE CHAPTER PLANE INCLUDING INTERSECTION OF TWO PLANES FOR THE MIDTERM EXAM. GOOD LUCK WITH THE EXAM. After the midterm exam, we will mention some situations between planes and lines we will see intersections of three planes and intersections of lines and planes and finally the specialized distance formulae and finish the chapter plane. If time permits, we will start the new chapter Locus Problems. We will also give circles determined various conditions. We will do some examples. I highly recommend you to read Matematik Dunyasi, 005 Yaz, Guz, Konikler. Bedia Akyar Moller
13 Analytic Geometry MAT WEEKLY PROGRAM 9-0 We talked about the chapter Plane. We have seen the equations of the planes which are perpendicular to the xy, xz and yz planes and also perpendicular to the coordinate axes. We have talked about the angle between two planes. I have also given the distance from a point to a plane (Please read the distance from a point to a line). We have seen the normal form of the equation (please read the normal form of a straight line). Although I have mentioned the situations between planes and lines I may again talk about the intersection of two planes and some examples by different methods (if it is necessary). We will see intersections of three planes and intersections of lines and planes and finally the specialized distance formulae and we will be finished the chapter plane. We will give the general form of a circle and also circles determined various conditions with aid of some examples. We are aiming to give intersections involving circles. I highly recommend you to read Matematik Dunyasi, 005 Yaz, Guz, Konikler. Do the following exercises: ) Find the vector formulation for the line of intersection of the planes P : x + y + 3z + 4 = 0 and P : x y + z = 0. ) Determine l P if l and P are given by l : (x, y, z) = (t +, t, 3t + ) and P : x y + z + = 0. 3) Find the distance from P 0 (3, 3, ) to the plane P : x + y z + 8 = 0. 4) Find an equation of the plane through (,,3) and perpendicular to the line x l = {(x, y, z) : = y+ = z }. 3 5) Find an equation of the plane through (,, 3), (,, ) and (-,,-3). x 6) Find an equation of the plane through (,, 3) and l = {(x, y, z) : = y+ = z }. 3 7) Find an equation of the plane determined by the lines l = {(x, y, z) : x + = y + = z} and l = {(t, t, 3t + 3) : t R}. 8) Determine the angle between the planes x+y z+7 = 0 and 3x y 6 = 0. 9) Find the cosine of the angle between the two planes with equations a) P : x + y + z = 0 and P : x y + z + 4 = 0, b) P 3 : x + y + z = 0 and P 4 : 4x y z 3 = 0, c) P 5 : x 6y + z = 0 and P 6 : x y + z 3 = 0.
14 0) Find a vector equation of the line of intersection of the planes in each case in the previous exercise. ) Find the equation of the plane parallel to the plane P : x 3y+7z = 0 and passing through (-, 3, 4). ) Find the equation of the plane perpendicular to the line joining (, 3, 5) and (4, 3, ) at the midpoint of these two points. 3) Determine α such that the following four planes P : x + y 3 = 0, P : 3x + 5z + 5 = 0, P 3 : x + y z 9 = 0 and P 4 : 3x + αy 5 = 0 which pass through the same point. Find the coordinates of this point. 4) Find a vector equation of the line through (,, 3) and in the direction of the vector u = (,, 3). 5) Find a vector equation of the line through the points (,, 3) and (,-, 3). 6) Find parametric equations of the line through (,,3) and parallel to the line through (,-,3) and (,-,4). 7) Find a symmetric equation of the line through (,,) and perpendicular to the two lines l = {(x, y, z) (t, t +, t)} and l = {(x, y, z) ( t, t, 3t + 3)}. 8) Find l l for given in the previous exercise. 9) Find the distance from P 0 (, 0, ) to the line l = {(x, y, z) (t, t +, t + )}. 0) Determine the locations (i.e. whether they are on the same side with the origin or not) of the points A(4,5,4), B(-5,8,-6) and C(3,5,4) with respect to the plane P : x y + 3z 7 = 0. ) Find the distance from (,, 3) to a) the line l = {(x, y, z) : x = y+ = z 3}. b) the plane P : x + y + z = 0. ) Find the distance from (, 0, ) to the line = {(x, y, z) : (x, y, z) = (t, t +, t + ), t R}. 3) Determine α such that the following four planes P : x + y 3 = 0, P : x + y z 9 = 0, P 3 : 3y + 5z + 5 = 0 and P 4 : 3x + αz 5 = 0 which pass through the same point. Find the coordinates of this point. 4) Determine l P if l and P are given by l = {(t +, t, 3t + ) : t R} and P = {(x, y, z) : x y + z + = 0}. 5) Determine P if and P have equations: a) l = {( t, t+, t ) : t R} and P = {(x, y, z) : x y +3z +4 = 0}. b) l = {(x, y, z) : x = y+ 0}. = z 3 } and P = {(x, y, z) : x + y + 3z 4 =
15 c) l = {(x, y, z) : x+6 3 = y = z } and P = {(x, y, z) : x y+z 4 = 0}. x 6) Given l = {(x, y, z) : = z }, find a line 3 l through (,, 3) such that l and l are skew. 7) Discuss the intersection of given planes with respect to the parameter λ. P : (λ + 5)x + (λ + 0)y + 3z 7 = 0, P : x + (λ + 7)y + z + 5 = 0. 8) Discuss the intersection of given planes with respect to the parameters a and b. P : x + y z + b = 0, P : x y + 3z = 0, P 3 : x + ay 6z + 0 = 0. 9) Find the distance between the lines l = {(x, y, z) : x + y z + 3 = 0, x y 4z+ = 0} and l = {(x, y, z) : 3x+y+5z = 0, y+z+ = 0}. 30) Find the equation of the circle tangent to the two axes and passing through the point (, 7). 3) Find the equation of the circle which is tangent to the y-axis, and passes through the point (, ) and the center of which is on the line x + y + 4 = 0. 3) Find the equation of the circle which passes through the points (, 4) and (4, 6) and whose center lies on the line 3x y + 0 = 0. 33) Show that for any value of θ the point (rcosθ, rsinθ) lies on the circle x + y = r. 34) Find an equation of the line which is tangent to the circle x + y + 4x 6y + 8 = 0 at the point ( 3, 5). 35) Find the equation of tangent line to each circle at the point indicated: a) x + y = 3, (, ). = y+ b) (x ) + (y + ) = 9, (, ). c) x + y + 4x 5y + 9 = 0, (, 3). 36) Find the equations of the tangents to the circle x + y = 6 drawn from the point ( 3, 7). 37) Find the equations of the tangents to (x + ) + (y ) = 9 with slope. 38) Find any points of intersection of curves a) x + y + 4x 6 = 0, x + 3y 0 = 0. b) x + y + 8x 6y = 0, x + y + = 0. 39) Find an equation of each tangent line drawn from the point (5, ) to the circle x + y = 3 by considering all possible methods. 40) Discuss the intersection of given lines and circles below, and if they intersect each other then find their intersection point.
16 a) l : x y 3 = 0, C : x + y 3x + y + 3 = 0. b) l : 3x y + 5 = 0, C : x + y + 4x y + 5 = 0. c) l : 3x y + = 0, C : x + y + x + y + 8 = 0. 4) Find any points of intersection of the given curves and an equation of radical axis, and draw a picture. a) x + y + x = 0, x + y y = 0. b) x + y + 8x 6y = 0, x + y + 3x y 4. c) x + y + 4x y = 0, x + y + 3x y 4 = 0. 4) Show that each of circles x + y = 9, x + y x + 7 = 0 and x + y 6x 8y + = 0 is tangent to other two. Do the common tangents meet in a point? If they do, find the point. 43) For which value of the parameter µ, the length of the tangent drawn from the point A(5, 4) to the circle C : x + y + µy = 0 is? 44) Show that the circles C : x + y 6x y + = 0 and C : x + y 4x + 4y + 6 = 0 intersect each other under right angle. 45) Discuss the intersection followings with respect to the parameter λ. a) l : λx y + = 0, C : x + y 0x + 4 = 0. b) C : x + y 3x + 8y 5 = 0, C : x + y + 6x y + λ = 0. Bedia Akyar Moller
17 Analytic Geometry MAT WEEKLY PROGRAM - I highly recommend you to read Matematik Dunyasi, 005 Yaz, Guz, Konikler. I am going to give another locus problem called Ellipse. I will give parametric equations of an Ellipse and the standard form of the equation of an ellipse. I am going to give last two locus problems called Hyperbola in the standard form and parabola, chord and the latus rectum of a Parabola. If time permits, after finishing the 4th chapter, I will start the next chapter and give Transformation of Coordinates. I am going to give translation of axes, rotation of axes and do some examples. Do the following exercises:. Find an equation of the indicated ellipse a) Foci ( 3, 0), a = 5. Also find the focal radii of a point for which x =. b) Foci (0, ), a = 4. c) Vertices ( 4, 0) and passing through the point (3, 5).. Find the coordinates of the foci and the vertices, and draw the graph of the equation a) 9x + 4y = 36. b) x + 4y = 6. c) x + 6y = Find an equation of the specified curves a) Of the locus of points the sum of whose distances from the points (0, ) is 6. b) Of the locus of points the sum of whose distances from the points (, 3) and (4, ) is Find an equation of the indicated hyperbolas.
18 a) Foci ( 5, 0), a = 3. Also find the focal radii of a point for which x = 6. b) Foci ( 3, 0), vertices (, 0). Also find the focal radii of a point for which x = 4. c) Foci (0, 4), b =. d) Foci (0, 5), ends of conjugate axis ( 3, 0). e) Vertices (0, 3) and passing through the point ( 3, 3). f) Foci ( 5, 0)and passing through the point ( 0 3, 4). g) Vertices (, 0) and asymptotes with slopes 3. h) Foci (0, 4) and (0, 0), and passing through the point (3, 4). 5. Find the foci, vertices, and asymptotes, and draw the graph of the equations: a) 6x y = 6. b) 8x y = 8. c) 4x 9y + 36 = Find an equation of the specified curves: a) Of the locus of points the difference of whose distances from the points (0, ) is. b) Of the locus of points the difference of whose distances from the points ( 4, 0) is 6. c) Of the locus of points the difference of whose distances from the points (, 3) and (4, ) is. 7. Find equations of asymptotes of the given hyperbolas in the form of single second degree equations: a) x 3y + 4 = 0. b) x y = 0. c) x 4 y 9 = Find an equation of the indicated parabolas. a) Directrix x =, focus (, 0). b) Directrix y = 4, focus (0, 4). c) Vertex (0, 0), directrix y = 4 3.
19 d) Vertex (0, 0), directrix x = 3 4. e) Vertex (0, 0), focus on the x axis, and passing through the point (8, 4). 9. Find the coordinates of the focus and equation of the directrix for the given parabolas and sketch the graphs. a) y = 6x. b) 3x + 4x = 0. c) x 8y = 0. d) 3x + y = Find the specified equations. a) Of the locus of points equidistant from the line y = 4 and the point 3 (0, 4). 3 b) Of the locus of points equidistant from the line x = 5 and the point ( 5, 0). c) Of the parabola with y = as directrix and (0, 4) as focus. d) Of the parabola with y = 3 as directrix and (, 0) as focus. e) Of the parabola with x + y + = 0 as directrix and (, 3) as focus. f) Of the parabola with 3x 4y = as directrix and (3, 0) as focus.
20 Analytic Geometry MAT WEEKLY PROGRAM 3 After finishing the 4th chapter, I have started the next chapter and given Transformation of Coordinates. We have already talked about translation of axes, rotation of axes and done some examples. Now I am going to start the new chapter Equations of Second degree. I am going to give classification of conics and conic sections. We are going to discuss all possible cases of conic sections. If time permits I will also talk about the family of conics. Do the following exercises:. a) Translate axes to the new origin (, 3) and reduce the equation of the curve x 4y 4x + 4y 36 = 0. b) For the equation x + 4xy + y = rotate the axes through θ = 45.. The x, y axes are translated to x, y axes with origin O (, 3). Find x, y coordinates of the point (4, 6). 3. Transform the equation 4x y 6x 7y 8 = 0 under the translation defined in Exercise. 4. Find x, y coordinates of the point (9, 5) under the rotation of axes through the obtuse angle θ for which tanθ = Transform the equation 3x 5y = under the rotation defined in Exercise 4. Bedia Akyar Moller
21 Analytic Geometry MAT WEEKLY PROGRAM 4 We have seen the chapter Equations of Second degree and given classification of conics and conic sections. We have discussed all possible cases of conic sections and I also talked about the family of conics.. Draw the graph of the following equations of second degree a) x + 4xy + 4y + x 4y + = 0. b) x + 4xy y + x + 4y = 0. c) 6x 4xy + 9y + 4y = 0. d) x + xy + y x y = 0.. Find the member of the family of lines through the intersection of x y+ = 0 and x + 3y 5 = 0 which passes through (, 5). 3. Find the equation of the line which passes through the point of intersection of x + y = 0 and x y + 7 = 0, and which is perpendicular to x + 6y 3 = What is the slope of the line joining the origin with the point of intersection of x 4y + = 0 and 3x + y + = 0? 5. Show that the three lines x y + 6 = 0, x + y 5 = 0 and x + y = 0 are concurrent (i.e. they meet in a common point). 6. Find k so that x + y + = 0, kx y + 3 = 0 and 4x 5y + k = 0 will be concurrent. 7. Find that member of the family of circles through the intersections of x + y 5x + y 4 = 0 and x + y + x 3y = 0 which passes through (, 5). 8. Determine the type of the following second degree equations depending on a parameter λ: a) x + y + 4x + λ = 0.
22 b) x 4y + y + λ = 0. c) x + 6x + (λ )y 4 = 0. d) 3λx + 3(λ )xy + (λ + )y + x y = 0. e) (λ )(x + y) x + y + = 0. f) x + λxy + λ(λ + )y + 4x = 0. g) λx + 4(λ )xy + (λ + 4)y + x y = Find the equation of the locus of the center of the conics 3x + (m + 8)xy + 4y x 8y = Find the locus of focus of the family of parabolas y = (m )x mx + m +.. Let C : x 4y + 4x y = 0 and C : x xy y + x + y = 0 be two conics. a) Find the equation of the family of conics passing through the intersection of the conics C and C. b) Discuss the family of conics for the parameter λ. c) Find the locus of the center of the conics in the family. d) Draw the graphs of the conic for λ =.. Let C : x + 6y 4x = 0 and C : x 4xy + 4y = 0 be two conics. a) Find the equation of the family of conics passing through the intersection of the conics C and C. b) Discuss the family of conics for the parameter λ. c) Find equation and draw the graph of locus of the center of the conics in the family. 3. Let l : x y = 0, l : x y = 0 and C : x + xy + y + y. a) Draw the graph of l, l and C. b) Find the equation of the family of conics passing through the intersection of the conic C and lines l, and l. c) Discuss the family of conics for the parameter λ. d) Find equation and draw the graph of locus of the center of the conics in the family.
23 4. Let C be a circle passing through the points (0, ), (, 0) and (, 0). Find the equation of the family conics passing through the points (, 0) and (, 0), and also tangent to the circle C at the point (0, ). 5. Find the equation of family of conics passing through the points O(0, 0), A(4, 0),B(0, 4) and whose normal lines at the point O(0, 0) bisect the line segment AB. Also, determine the types of members of the family depending on a parameter λ. 6. Find the equation of family of conics passing through the points O(0, 0), A(, 0),B(0, ) and whose tangent lines at the point O(0, 0) passing through C( 4, ). Also, determine the types of members of the family depending 3 3 on a parameter λ. Bedia Akyar Moller
Analytic Geometry MAT 1035
Analytic Geometry MAT 035 5.09.04 WEEKLY PROGRAM - The first week of the semester, we will introduce the course and given a brief outline. We continue with vectors in R n and some operations including
More informationMAT1035 Analytic Geometry
MAT1035 Analytic Geometry Lecture Notes R.A. Sabri Kaan Gürbüzer Dokuz Eylül University 2016 2 Contents 1 Review of Trigonometry 5 2 Polar Coordinates 7 3 Vectors in R n 9 3.1 Located Vectors..............................................
More informationUnit 8. ANALYTIC GEOMETRY.
Unit 8. ANALYTIC GEOMETRY. 1. VECTORS IN THE PLANE A vector is a line segment running from point A (tail) to point B (head). 1.1 DIRECTION OF A VECTOR The direction of a vector is the direction of the
More informationTARGET : JEE 2013 SCORE. JEE (Advanced) Home Assignment # 03. Kota Chandigarh Ahmedabad
TARGT : J 01 SCOR J (Advanced) Home Assignment # 0 Kota Chandigarh Ahmedabad J-Mathematics HOM ASSIGNMNT # 0 STRAIGHT OBJCTIV TYP 1. If x + y = 0 is a tangent at the vertex of a parabola and x + y 7 =
More informationSenior Math Circles February 18, 2009 Conics III
University of Waterloo Faculty of Mathematics Senior Math Circles February 18, 2009 Conics III Centre for Education in Mathematics and Computing Eccentricity of Conics Fix a point F called the focus, a
More informationThe Distance Formula. The Midpoint Formula
Math 120 Intermediate Algebra Sec 9.1: Distance Midpoint Formulas The Distance Formula The distance between two points P 1 = (x 1, y 1 ) P 2 = (x 1, y 1 ), denoted by d(p 1, P 2 ), is d(p 1, P 2 ) = (x
More informationDistance and Midpoint Formula 7.1
Distance and Midpoint Formula 7.1 Distance Formula d ( x - x ) ( y - y ) 1 1 Example 1 Find the distance between the points (4, 4) and (-6, -). Example Find the value of a to make the distance = 10 units
More informationQUESTION BANK ON STRAIGHT LINE AND CIRCLE
QUESTION BANK ON STRAIGHT LINE AND CIRCLE Select the correct alternative : (Only one is correct) Q. If the lines x + y + = 0 ; 4x + y + 4 = 0 and x + αy + β = 0, where α + β =, are concurrent then α =,
More informationy 1 x 1 ) 2 + (y 2 ) 2 A circle is a set of points P in a plane that are equidistant from a fixed point, called the center.
Ch 12. Conic Sections Circles, Parabolas, Ellipses & Hyperbolas The formulas for the conic sections are derived by using the distance formula, which was derived from the Pythagorean Theorem. If you know
More informationQUESTION BANK ON. CONIC SECTION (Parabola, Ellipse & Hyperbola)
QUESTION BANK ON CONIC SECTION (Parabola, Ellipse & Hyperbola) Question bank on Parabola, Ellipse & Hyperbola Select the correct alternative : (Only one is correct) Q. Two mutually perpendicular tangents
More informationGive a geometric description of the set of points in space whose coordinates satisfy the given pair of equations.
1. Give a geometric description of the set of points in space whose coordinates satisfy the given pair of equations. x + y = 5, z = 4 Choose the correct description. A. The circle with center (0,0, 4)
More informationchapter 1 vector geometry solutions V Consider the parallelogram shown alongside. Which of the following statements are true?
chapter vector geometry solutions V. Exercise A. For the shape shown, find a single vector which is equal to a)!!! " AB + BC AC b)! AD!!! " + DB AB c)! AC + CD AD d)! BC + CD!!! " + DA BA e) CD!!! " "
More informationConic section. Ans: c. Ans: a. Ans: c. Episode:43 Faculty: Prof. A. NAGARAJ. 1. A circle
Episode:43 Faculty: Prof. A. NAGARAJ Conic section 1. A circle gx fy c 0 is said to be imaginary circle if a) g + f = c b) g + f > c c) g + f < c d) g = f. If (1,-3) is the centre of the circle x y ax
More informationOHSx XM521 Multivariable Differential Calculus: Homework Solutions 13.1
OHSx XM521 Multivariable Differential Calculus: Homework Solutions 13.1 (37) If a bug walks on the sphere x 2 + y 2 + z 2 + 2x 2y 4z 3 = 0 how close and how far can it get from the origin? Solution: Complete
More information( 1 ) Find the co-ordinates of the focus, length of the latus-rectum and equation of the directrix of the parabola x 2 = - 8y.
PROBLEMS 04 - PARABOLA Page 1 ( 1 ) Find the co-ordinates of the focus, length of the latus-rectum and equation of the directrix of the parabola x - 8. [ Ans: ( 0, - ), 8, ] ( ) If the line 3x 4 k 0 is
More information3. A( 2,0) and B(6, -2), find M 4. A( 3, 7) and M(4,-3), find B. 5. M(4, -9) and B( -10, 11) find A 6. B(4, 8) and M(-2, 5), find A
Midpoint and Distance Formula Class Work M is the midpoint of A and B. Use the given information to find the missing point. 1. A(4, 2) and B(3, -8), find M 2. A(5, 7) and B( -2, -9), find M 3. A( 2,0)
More informationMathematics. Single Correct Questions
Mathematics Single Correct Questions +4 1.00 1. If and then 2. The number of solutions of, in the interval is : 3. If then equals : 4. A plane bisects the line segment joining the points and at right angles.
More informationPortable Assisted Study Sequence ALGEBRA IIB
SCOPE This course is divided into two semesters of study (A & B) comprised of five units each. Each unit teaches concepts and strategies recommended for intermediate algebra students. The second half of
More information9.1 Circles and Parabolas. Copyright Cengage Learning. All rights reserved.
9.1 Circles and Parabolas Copyright Cengage Learning. All rights reserved. What You Should Learn Recognize a conic as the intersection of a plane and a double-napped cone. Write equations of circles in
More informationParabola. The fixed point is called the focus and it is denoted by S. A (0, 0), S (a, 0) and P (x 1, y 1 ) PM=NZ=NA+AZ= x 1 + a
: Conic: The locus of a point moving on a plane such that its distance from a fixed point and a fixed straight line in the plane are in a constant ratio é, is called a conic. The fixed point is called
More informationMATH-1420 Review Concepts (Haugen)
MATH-40 Review Concepts (Haugen) Unit : Equations, Inequalities, Functions, and Graphs Rational Expressions Determine the domain of a rational expression Simplify rational expressions -factor and then
More informationGrade XI Mathematics
Grade XI Mathematics Exam Preparation Booklet Chapter Wise - Important Questions and Solutions #GrowWithGreen Questions Sets Q1. For two disjoint sets A and B, if n [P ( A B )] = 32 and n [P ( A B )] =
More informationTS EAMCET 2016 SYLLABUS ENGINEERING STREAM
TS EAMCET 2016 SYLLABUS ENGINEERING STREAM Subject: MATHEMATICS 1) ALGEBRA : a) Functions: Types of functions Definitions - Inverse functions and Theorems - Domain, Range, Inverse of real valued functions.
More informationQuantities which have only magnitude are called scalars. Quantities which have magnitude and direction are called vectors.
Vectors summary Quantities which have only magnitude are called scalars. Quantities which have magnitude and direction are called vectors. AB is the position vector of B relative to A and is the vector
More informationKEMATH1 Calculus for Chemistry and Biochemistry Students. Francis Joseph H. Campeña, De La Salle University Manila
KEMATH1 Calculus for Chemistry and Biochemistry Students Francis Joseph H Campeña, De La Salle University Manila February 9, 2015 Contents 1 Conic Sections 2 11 A review of the coordinate system 2 12 Conic
More informationKing Fahd University of Petroleum and Minerals Prep-Year Math Program Math (001) - Term 181 Recitation (1.1)
Recitation (1.1) Question 1: Find a point on the y-axis that is equidistant from the points (5, 5) and (1, 1) Question 2: Find the distance between the points P(2 x, 7 x) and Q( 2 x, 4 x) where x 0. Question
More informationExercise. Exercise 1.1. MA112 Section : Prepared by Dr.Archara Pacheenburawana 1
MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 1 Exercise Exercise 1.1 1 8 Find the vertex, focus, and directrix of the parabola and sketch its graph. 1. x = 2y 2 2. 4y +x 2 = 0 3. 4x 2 =
More informationPreCalculus Honors Curriculum Pacing Guide First Half of Semester
Unit 1 Introduction to Trigonometry (9 days) First Half of PC.FT.1 PC.FT.2 PC.FT.2a PC.FT.2b PC.FT.3 PC.FT.4 PC.FT.8 PC.GCI.5 Understand that the radian measure of an angle is the length of the arc on
More informationGAT-UGTP-2018 Page 1 of 5
SECTION A: MATHEMATICS UNIT 1 SETS, RELATIONS AND FUNCTIONS: Sets and their representation, Union, Intersection and compliment of sets, and their algebraic properties, power set, Relation, Types of relation,
More information8. Find r a! r b. a) r a = [3, 2, 7], r b = [ 1, 4, 5] b) r a = [ 5, 6, 7], r b = [2, 7, 4]
Chapter 8 Prerequisite Skills BLM 8-1.. Linear Relations 1. Make a table of values and graph each linear function a) y = 2x b) y = x + 5 c) 2x + 6y = 12 d) x + 7y = 21 2. Find the x- and y-intercepts of
More information3. A( 2,0) and B(6, -2), find M 4. A( 3, 7) and M(4,-3), find B. 5. M(4, -9) and B( -10, 11) find A 6. B(4, 8) and M(-2, 5), find A
Midpoint and Distance Formula Class Work M is the midpoint of A and B. Use the given information to find the missing point. 1. A(, 2) and B(3, -8), find M 2. A(5, 7) and B( -2, -), find M (3. 5, 3) (1.
More informationCIRCLES: #1. What is an equation of the circle at the origin and radius 12?
1 Pre-AP Algebra II Chapter 10 Test Review Standards/Goals: E.3.a.: I can identify conic sections (parabola, circle, ellipse, hyperbola) from their equations in standard form. E.3.b.: I can graph circles
More information1. Matrices and Determinants
Important Questions 1. Matrices and Determinants Ex.1.1 (2) x 3x y Find the values of x, y, z if 2x + z 3y w = 0 7 3 2a Ex 1.1 (3) 2x 3x y If 2x + z 3y w = 3 2 find x, y, z, w 4 7 Ex 1.1 (13) 3 7 3 2 Find
More informationConic Sections. Geometry - Conics ~1~ NJCTL.org. Write the following equations in standard form.
Conic Sections Midpoint and Distance Formula M is the midpoint of A and B. Use the given information to find the missing point. 1. A(, 2) and B(3, -), find M 2. A(5, 7) and B( -2, -), find M 3. A( 2,0)
More informationAlgebra 2A Unit 1 Week 1 Day Activity Unit 1 Week 2 Day Activity Unit 1 Week 3 Day Activity Unit 2 Week 1 Day Activity
Algebra 2A Unit 1 Week 1 1 Pretest Unit 1 2 Evaluating Rational Expressions 3 Restrictions on Rational Expressions 4 Equivalent Forms of Rational Expressions 5 Simplifying Rational Expressions Unit 1 Week
More informationChapter 13: Vectors and the Geometry of Space
Chapter 13: Vectors and the Geometry of Space 13.1 3-Dimensional Coordinate System 13.2 Vectors 13.3 The Dot Product 13.4 The Cross Product 13.5 Equations of Lines and Planes 13.6 Cylinders and Quadratic
More informationChapter 13: Vectors and the Geometry of Space
Chapter 13: Vectors and the Geometry of Space 13.1 3-Dimensional Coordinate System 13.2 Vectors 13.3 The Dot Product 13.4 The Cross Product 13.5 Equations of Lines and Planes 13.6 Cylinders and Quadratic
More informationTARGET QUARTERLY MATHS MATERIAL
Adyar Adambakkam Pallavaram Pammal Chromepet Now also at SELAIYUR TARGET QUARTERLY MATHS MATERIAL Achievement through HARDWORK Improvement through INNOVATION Target Centum Practising Package +2 GENERAL
More informationHomework Assignments Math /02 Fall 2014
Homework Assignments Math 119-01/02 Fall 2014 Assignment 1 Due date : Friday, September 5 6th Edition Problem Set Section 6.1, Page 178: #1, 2, 3, 4, 5, 6. Section 6.2, Page 185: #1, 2, 3, 5, 6, 8, 10-14,
More informationPARABOLA. AIEEE Syllabus. Total No. of questions in Parabola are: Solved examples Level # Level # Level # Level # 4..
PRBOL IEEE yllabus 1. Definition. Terms related to Parabola 3. tandard form of Equation of Parabola 4. Reduction to standard Equation 5. General Equation of a Parabola 6. Equation of Parabola when its
More informationConic Sections Session 2: Ellipse
Conic Sections Session 2: Ellipse Toh Pee Choon NIE Oct 2017 Toh Pee Choon (NIE) Session 2: Ellipse Oct 2017 1 / 24 Introduction Problem 2.1 Let A, F 1 and F 2 be three points that form a triangle F 2
More informationax 2 + bx + c = 0 where
Chapter P Prerequisites Section P.1 Real Numbers Real numbers The set of numbers formed by joining the set of rational numbers and the set of irrational numbers. Real number line A line used to graphically
More informationCalculus III. George Voutsadakis 1. LSSU Math 251. Lake Superior State University. 1 Mathematics and Computer Science
Calculus III George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 251 George Voutsadakis (LSSU) Calculus III January 2016 1 / 76 Outline 1 Parametric Equations,
More informationJanuary 21, 2018 Math 9. Geometry. The method of coordinates (continued). Ellipse. Hyperbola. Parabola.
January 21, 2018 Math 9 Ellipse Geometry The method of coordinates (continued) Ellipse Hyperbola Parabola Definition An ellipse is a locus of points, such that the sum of the distances from point on the
More informationALGEBRA 2 X. Final Exam. Review Packet
ALGEBRA X Final Exam Review Packet Multiple Choice Match: 1) x + y = r a) equation of a line ) x = 5y 4y+ b) equation of a hyperbola ) 4) x y + = 1 64 9 c) equation of a parabola x y = 1 4 49 d) equation
More informationPre-Calculus & Trigonometry Scope and Sequence
WHCSD Scope and Sequence Pre-Calculus/ 2017-2018 Pre-Calculus & Scope and Sequence Course Overview and Timing This section is to help you see the flow of the unit/topics across the entire school year.
More informationCircles. Example 2: Write an equation for a circle if the enpoints of a diameter are at ( 4,5) and (6, 3).
Conics Unit Ch. 8 Circles Equations of Circles The equation of a circle with center ( hk, ) and radius r units is ( x h) ( y k) r. Example 1: Write an equation of circle with center (8, 3) and radius 6.
More informationMathematics Precalculus: Academic Unit 7: Conics
Understandings Questions Knowledge Vocabulary Skills Conics are models of real-life situations. Conics have many reflective properties that are used in every day situations Conics work can be simplified
More informationDetailed objectives are given in each of the sections listed below. 1. Cartesian Space Coordinates. 2. Displacements, Forces, Velocities and Vectors
Unit 1 Vectors In this unit, we introduce vectors, vector operations, and equations of lines and planes. Note: Unit 1 is based on Chapter 12 of the textbook, Salas and Hille s Calculus: Several Variables,
More informationMidterm Review Packet. Geometry: Midterm Multiple Choice Practice
: Midterm Multiple Choice Practice 1. In the diagram below, a square is graphed in the coordinate plane. A reflection over which line does not carry the square onto itself? (1) (2) (3) (4) 2. A sequence
More informationHomework Assignments Math /02 Fall 2017
Homework Assignments Math 119-01/02 Fall 2017 Assignment 1 Due date : Wednesday, August 30 Section 6.1, Page 178: #1, 2, 3, 4, 5, 6. Section 6.2, Page 185: #1, 2, 3, 5, 6, 8, 10-14, 16, 17, 18, 20, 22,
More informationPARAMETRIC EQUATIONS AND POLAR COORDINATES
10 PARAMETRIC EQUATIONS AND POLAR COORDINATES PARAMETRIC EQUATIONS & POLAR COORDINATES 10.5 Conic Sections In this section, we will learn: How to derive standard equations for conic sections. CONIC SECTIONS
More informationMath 3c Solutions: Exam 1 Fall Graph by eliiminating the parameter; be sure to write the equation you get when you eliminate the parameter.
Math c Solutions: Exam 1 Fall 16 1. Graph by eliiminating the parameter; be sure to write the equation you get when you eliminate the parameter. x tan t x tan t y sec t y sec t t π 4 To eliminate the parameter,
More information(D) (A) Q.3 To which of the following circles, the line y x + 3 = 0 is normal at the point ? 2 (A) 2
CIRCLE [STRAIGHT OBJECTIVE TYPE] Q. The line x y + = 0 is tangent to the circle at the point (, 5) and the centre of the circles lies on x y = 4. The radius of the circle is (A) 3 5 (B) 5 3 (C) 5 (D) 5
More informationConic Sections Session 3: Hyperbola
Conic Sections Session 3: Hyperbola Toh Pee Choon NIE Oct 2017 Toh Pee Choon (NIE) Session 3: Hyperbola Oct 2017 1 / 16 Problem 3.1 1 Recall that an ellipse is defined as the locus of points P such that
More informationSection 8.1 Vector and Parametric Equations of a Line in
Section 8.1 Vector and Parametric Equations of a Line in R 2 In this section, we begin with a discussion about how to find the vector and parametric equations of a line in R 2. To find the vector and parametric
More informationTime : 3 hours 02 - Mathematics - July 2006 Marks : 100 Pg - 1 Instructions : S E CT I O N - A
Time : 3 hours 0 Mathematics July 006 Marks : 00 Pg Instructions :. Answer all questions.. Write your answers according to the instructions given below with the questions. 3. Begin each section on a new
More information2. (i) Find the equation of the circle which passes through ( 7, 1) and has centre ( 4, 3).
Circle 1. (i) Find the equation of the circle with centre ( 7, 3) and of radius 10. (ii) Find the centre of the circle 2x 2 + 2y 2 + 6x + 8y 1 = 0 (iii) What is the radius of the circle 3x 2 + 3y 2 + 5x
More informationPart (1) Second : Trigonometry. Tan
Part (1) Second : Trigonometry (1) Complete the following table : The angle Ratio 42 12 \ Sin 0.3214 Cas 0.5321 Tan 2.0625 (2) Complete the following : 1) 46 36 \ 24 \\ =. In degrees. 2) 44.125 = in degrees,
More informationSaxon Advanced Mathematics Scope and Sequence
hmhco.com Saxon Advanced Mathematics Scope and Sequence Foundations Calculator Perform two-variable analysis Use graphing calculators Find roots of equations Solve systems of equations Exponentials and
More informationPrecalculus. Precalculus Higher Mathematics Courses 85
Precalculus Precalculus combines the trigonometric, geometric, and algebraic techniques needed to prepare students for the study of calculus, and strengthens students conceptual understanding of problems
More informationCommon Core Edition Table of Contents
Common Core Edition Table of Contents ALGEBRA 1 Chapter 1 Foundations for Algebra 1-1 Variables and Expressions 1-2 Order of Operations and Evaluating Expressions 1-3 Real Numbers and the Number Line 1-4
More information11.1 Three-Dimensional Coordinate System
11.1 Three-Dimensional Coordinate System In three dimensions, a point has three coordinates: (x,y,z). The normal orientation of the x, y, and z-axes is shown below. The three axes divide the region into
More informationExam. There are 6 problems. Your 5 best answers count. Please pay attention to the presentation of your work! Best 5
Department of Mathematical Sciences Instructor: Daiva Pucinskaite Calculus III June, 06 Name: Exam There are 6 problems. Your 5 best answers count. Please pay attention to the presentation of your work!
More informationIYGB. Special Paper U. Time: 3 hours 30 minutes. Created by T. Madas. Created by T. Madas
IYGB Special Paper U Time: 3 hours 30 minutes Candidates may NOT use any calculator Information for Candidates This practice paper follows the Advanced Level Mathematics Core Syllabus Booklets of Mathematical
More informationMATH 255 Applied Honors Calculus III Winter Midterm 1 Review Solutions
MATH 55 Applied Honors Calculus III Winter 11 Midterm 1 Review Solutions 11.1: #19 Particle starts at point ( 1,, traces out a semicircle in the counterclockwise direction, ending at the point (1,. 11.1:
More informationy mx 25m 25 4 circle. Then the perpendicular distance of tangent from the centre (0, 0) is the radius. Since tangent
Mathematics. The sides AB, BC and CA of ABC have, 4 and 5 interior points respectively on them as shown in the figure. The number of triangles that can be formed using these interior points is () 80 ()
More informationGLOBAL TALENT SEARCH EXAMINATIONS (GTSE) CLASS -XI
GLOBAL TALENT SEARCH EXAMINATIONS (GTSE) Date: rd November 008 CLASS -XI MATHEMATICS Max Marks: 80 Time: :0 to :5 a.m. General Instructions: (Read Instructions carefully). All questions are compulsory.
More informationMath 1012 Precalculus Functions Final Exam Review Page 1 of 5
Math 1012 Precalculus Functions Final Exam Review Page 1 of 5 Final Exam Preparation The final exam will be 10 questions long, some with two parts. Material for the final can be drawn from anything we
More informationMAT 1339-S14 Class 10 & 11
MAT 1339-S14 Class 10 & 11 August 7 & 11, 2014 Contents 8 Lines and Planes 1 8.1 Equations of Lines in Two-Space and Three-Space............ 1 8.2 Equations of Planes........................... 5 8.3 Properties
More informationb = 2, c = 3, we get x = 0.3 for the positive root. Ans. (D) x 2-2x - 8 < 0, or (x - 4)(x + 2) < 0, Therefore -2 < x < 4 Ans. (C)
SAT II - Math Level 2 Test #02 Solution 1. The positive zero of y = x 2 + 2x is, to the nearest tenth, equal to (A) 0.8 (B) 0.7 + 1.1i (C) 0.7 (D) 0.3 (E) 2.2 ± Using Quadratic formula, x =, with a = 1,
More information= 0 1 (3 4 ) 1 (4 4) + 1 (4 3) = = + 1 = 0 = 1 = ± 1 ]
STRAIGHT LINE [STRAIGHT OBJECTIVE TYPE] Q. If the lines x + y + = 0 ; x + y + = 0 and x + y + = 0, where + =, are concurrent then (A) =, = (B) =, = ± (C) =, = ± (D*) = ±, = [Sol. Lines are x + y + = 0
More informationCongruence Axioms. Data Required for Solving Oblique Triangles
Math 335 Trigonometry Sec 7.1: Oblique Triangles and the Law of Sines In section 2.4, we solved right triangles. We now extend the concept to all triangles. Congruence Axioms Side-Angle-Side SAS Angle-Side-Angle
More informationALGEBRA II Grades 9-12
Summer 2015 Units: 10 high school credits UC Requirement Category: c General Description: ALGEBRA II Grades 9-12 Algebra II is a course which further develops the concepts learned in Algebra I. It will
More informationMath III Curriculum Map
6 weeks Unit Unit Focus Common Core Math Standards 1 Rational and Irrational Numbers N-RN.3. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an
More information1. The positive zero of y = x 2 + 2x 3/5 is, to the nearest tenth, equal to
SAT II - Math Level Test #0 Solution SAT II - Math Level Test No. 1. The positive zero of y = x + x 3/5 is, to the nearest tenth, equal to (A) 0.8 (B) 0.7 + 1.1i (C) 0.7 (D) 0.3 (E). 3 b b 4ac Using Quadratic
More informationWA State Common Core Standards - Mathematics
Number & Quantity The Real Number System Extend the properties of exponents to rational exponents. 1. Explain how the definition of the meaning of rational exponents follows from extending the properties
More informationPRECALCULUS. Changes to the original 2010 COS is in red. If it is red and crossed out, it has been moved to another course.
PRECALCULUS Precalculus is a course designed for students who have successfully completed the Algebra II With Trigonometry course. This course is considered to be a prerequisite for success in calculus
More informationQ1. If (1, 2) lies on the circle. x 2 + y 2 + 2gx + 2fy + c = 0. which is concentric with the circle x 2 + y 2 +4x + 2y 5 = 0 then c =
Q1. If (1, 2) lies on the circle x 2 + y 2 + 2gx + 2fy + c = 0 which is concentric with the circle x 2 + y 2 +4x + 2y 5 = 0 then c = a) 11 b) -13 c) 24 d) 100 Solution: Any circle concentric with x 2 +
More informationMaths Higher Prelim Content
Maths Higher Prelim Content Straight Line Gradient of a line A(x 1, y 1 ), B(x 2, y 2 ), Gradient of AB m AB = y 2 y1 x 2 x 1 m = tanθ where θ is the angle the line makes with the positive direction of
More information30. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. [F-TF]
Pre-Calculus Curriculum Map (Revised April 2015) Unit Content Standard Unit 1 Unit Circle 29. (+) Use special triangles to determine geometrically the values of sine, cosine, and tangent for and use the
More informationTheorem 1.2 (Converse of Pythagoras theorem). If the lengths of the sides of ABC satisfy a 2 + b 2 = c 2, then the triangle has a right angle at C.
hapter 1 Some asic Theorems 1.1 The ythagorean Theorem Theorem 1.1 (ythagoras). The lengths a b < c of the sides of a right triangle satisfy the relation a + b = c. roof. b a a 3 b b 4 b a b 4 1 a a 3
More informationSTAAR STANDARDS ALGEBRA I ALGEBRA II GEOMETRY
STANDARDS ALGEBRA I ALGEBRA II GEOMETRY STANDARDS ALGEBRA I TEKS Snapshot Algebra I (New TEKS 2015-16) Mathematical Process Standards A.1 Mathematical process standards. The student uses mathematical processes
More information4.Let A be a matrix such that A. is a scalar matrix and Then equals :
1.Consider the following two binary relations on the set A={a, b, c} : R1={(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)} and R2={(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)}. Then : both R1
More informationTWO THEOREMS ON THE FOCUS-SHARING ELLIPSES: A THREE-DIMENSIONAL VIEW
TWO THEOREMS ON THE FOCUS-SHARING ELLIPSES: A THREE-DIMENSIONAL VIEW ILYA I. BOGDANOV Abstract. Consider three ellipses each two of which share a common focus. The radical axes of the pairs of these ellipses
More informationNozha Directorate of Education Form : 2 nd Prep. Nozha Language Schools Ismailia Road Branch
Cairo Governorate Department : Maths Nozha Directorate of Education Form : 2 nd Prep. Nozha Language Schools Sheet Ismailia Road Branch Sheet ( 1) 1-Complete 1. in the parallelogram, each two opposite
More informationSYSTEM OF CIRCLES OBJECTIVES (a) Touch each other internally (b) Touch each other externally
SYSTEM OF CIRCLES OBJECTIVES. A circle passes through (0, 0) and (, 0) and touches the circle x + y = 9, then the centre of circle is (a) (c) 3,, (b) (d) 3,, ±. The equation of the circle having its centre
More informationCRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 2012, Brooks/Cole
More informationAlg. (( Sheet 1 )) [1] Complete : 1) =.. 3) =. 4) 3 a 3 =.. 5) X 3 = 64 then X =. 6) 3 X 6 =... 7) 3
Cairo Governorate Department : Maths Nozha Directorate of Education Form : 2 nd Prep. Nozha Language Schools Sheet Ismailia Road Branch [1] Complete : 1) 3 216 =.. Alg. (( Sheet 1 )) 1 8 2) 3 ( ) 2 =..
More informationPage 1
Pacing Chart Unit Week Day CCSS Standards Objective I Can Statements 121 CCSS.MATH.CONTENT.HSG.C.A.1 Prove that all circles are similar. Prove that all circles are similar. I can prove that all circles
More informationChapter 1 Analytic geometry in the plane
3110 General Mathematics 1 31 10 General Mathematics For the students from Pharmaceutical Faculty 1/004 Instructor: Dr Wattana Toutip (ดร.ว ฒนา เถาว ท พย ) Chapter 1 Analytic geometry in the plane Overview:
More information0811ge. Geometry Regents Exam BC, AT = 5, TB = 7, and AV = 10.
0811ge 1 The statement "x is a multiple of 3, and x is an even integer" is true when x is equal to 1) 9 2) 8 3) 3 4) 6 2 In the diagram below, ABC XYZ. 4 Pentagon PQRST has PQ parallel to TS. After a translation
More informationChapter 2. The laws of sines and cosines. 2.1 The law of sines. Theorem 2.1 (The law of sines). Let R denote the circumradius of a triangle ABC.
hapter 2 The laws of sines and cosines 2.1 The law of sines Theorem 2.1 (The law of sines). Let R denote the circumradius of a triangle. 2R = a sin α = b sin β = c sin γ. α O O α as Since the area of a
More informationChapter 9. Conic Sections and Analytic Geometry. 9.2 The Hyperbola. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter 9 Conic Sections and Analytic Geometry 9. The Hyperbola Copyright 014, 010, 007 Pearson Education, Inc. 1 Objectives: Locate a hyperbola s vertices and foci. Write equations of hyperbolas in standard
More informationIMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS
` KUKATPALLY CENTRE IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 57 Ph: 4-646113
More informationCUMBERLAND COUNTY SCHOOL DISTRICT BENCHMARK ASSESSMENT CURRICULUM PACING GUIDE
CUMBERLAND COUNTY SCHOOL DISTRICT BENCHMARK ASSESSMENT CURRICULUM PACING GUIDE School: CCHS Subject: Algebra II Grade: 10 th Grade Benchmark Assessment 1 Instructional Timeline: 1 st Nine Weeks Topic(s):
More information3. Interpret the graph of x = 1 in the contexts of (a) a number line (b) 2-space (c) 3-space
MA2: Prepared by Dr. Archara Pacheenburawana Exercise Chapter 3 Exercise 3.. A cube of side 4 has its geometric center at the origin and its faces parallel to the coordinate planes. Sketch the cube and
More informationA plane in which each point is identified with a ordered pair of real numbers (x,y) is called a coordinate (or Cartesian) plane.
Coordinate Geometry Rene Descartes, considered the father of modern philosophy (Cogito ergo sum), also had a great influence on mathematics. He and Fermat corresponded regularly and as a result of their
More information130 Important Questions for XI
130 Important Questions for XI E T V A 1 130 Important Questions for XI PREFACE Have you ever seen a plane taking off from a runway and going up and up, and crossing the clouds but just think again that
More informationPre-Calculus EOC Review 2016
Pre-Calculus EOC Review 2016 Name The Exam 50 questions, multiple choice, paper and pencil. I. Limits 8 questions a. (1) decide if a function is continuous at a point b. (1) understand continuity in terms
More information