Conic Sections Session 2: Ellipse
|
|
- Brett Fletcher
- 6 years ago
- Views:
Transcription
1 Conic Sections Session 2: Ellipse Toh Pee Choon NIE Oct 2017 Toh Pee Choon (NIE) Session 2: Ellipse Oct / 24
2 Introduction Problem 2.1 Let A, F 1 and F 2 be three points that form a triangle F 2 F 1 A. Find all points P such that the triangle F 2 F 1 P has the same area as F 2 F 1 A. Can you describe the locus of P? The locus of P is l: the line through A and parallel to the line F 2 F 1. (Is that all?) l the reflection of l about F 2 F 1 should also be included. Do we have a more elegant way to describe the locus? Define d as the line F 2 F 1 and for any point X recall that Xd denotes the perpendicular distance from X to d. The required locus is {P : Pd = Ad = k} for some real constant k. Toh Pee Choon (NIE) Session 2: Ellipse Oct / 24
3 Introduction Problem 2.2 Let A, F 1 and F 2 be three points that form a triangle F 2 F 1 A. Find all points P such that the triangle F 2 F 1 P has the same perimeter as F 2 F 1 A. Can you describe the locus of P? The locus of P is {P : PF 2 + PF 1 = AF 2 + AF 1 = k} for some real constant k. Can you describe the locus in geometric terms? It is an ellipse (passing through A) with F 2 and F 1 as its foci. Toh Pee Choon (NIE) Session 2: Ellipse Oct / 24
4 Definition of Ellipse Definition 2.3 (Geometric) An ellipse is the locus of points whose sum of distances to the foci F 1 and F 2 is a constant, i.e. {P : PF 1 + PF 2 = k}. B P A F 2 F 1 Toh Pee Choon (NIE) Session 2: Ellipse Oct / 24
5 Definition 2.4 An ellipse in standard position is defined by x 2 a 2 + y 2 = 1, where a b > 0. b2 The major and minor axes are segments through the center that correspond respectively to the largest and smallest distance between antipodal points on the ellipse. The vertices are the points of intersection of the ellipse and the major/minor axis. i.e. vertices are A = (a, 0), B = ( a, 0), C = (0, b) and D = (0, b). AB is the major axis and CD is the minor axis. (The analogous definitions hold when b > a.) C B (0, 0) A D Toh Pee Choon (NIE) Session 2: Ellipse Oct / 24
6 Problem 2.5 (Relating the Cartesian and Geometric definitions) An ellipse with major and minor axes on the x and y axes respectively and vertices (a, 0) and (0, b) is defined by PF 1 + PF 2 = k. Find the value of k and coordinates of F 1 and F 2. Hence calculate the focal length F 1 F 2. The coordinates of F 1 and F 2 must be (c, 0) and ( c, 0) respectively. When P = (a, 0), AF 1 + AF 2 = a c + (a + c) = 2a. When P = (0, b), by Pythagoras Theorem, BF 1 + BF 2 = 2 c 2 + b 2 = 2a. Hence k = 2a and c = a 2 b 2. Finally the focal length is 2 a 2 b 2. Toh Pee Choon (NIE) Session 2: Ellipse Oct / 24
7 Problem 2.6 (Geometric = Cartesian) Derive the Cartesian equation from the Geometric definition. Since c = a 2 b 2, PF 1 + PF 2 = 2a, we have (x c) 2 + y 2 + (x + c) 2 + y 2 = 2a = (x + c) 2 + y 2 = (2a ) 2 (x c) 2 + y 2 = 4a 2 4a (x c) 2 + y 2 + (x c) 2 + y 2 = 4xc 4a 2 = 4a (x c) 2 + y 2 = ( a 2 cx ) 2 = a 2 (x c) 2 + a 2 y 2 = a 4 + c 2 x 2 = a 2 x 2 + a 2 c 2 + a 2 y 2 = a 2 (a 2 c 2 ) = (a 2 c 2 )x 2 + a 2 y 2 = a 2 b 2 = b 2 x 2 + a 2 y 2. Toh Pee Choon (NIE) Session 2: Ellipse Oct / 24
8 Exercise 2.7 Let x 2 a 2 + y 2 b 2 = 1, a b > 0 represent an ellipse in standard position and F be a focus. Find the length of the chord parallel to minor axis and passing through F. This chord is called a latus rectum. Consider F 1 = (c, 0) where c = a 2 b 2 So y 2 = 1 a2 b 2 b 2 a 2 i.e. y = ± b2 a. So the latus rectum is of length 2b2 a. Toh Pee Choon (NIE) Session 2: Ellipse Oct / 24
9 Exercise 2.8 If the major axis of an ellipse given by the equation x t + y 2 t 2 2 = 1 lies on the y-axis, find the range of values of t. We require 4 + t > 0 = t > 4 and t 2 2 > 0 = t < 2 or t > 2. Furthermore if the major axis is on the y-axis, we need t t. i.e. t 2 t 6 0 = t 3 or t 2. Combining the two gives t 3 or 4 < t 2. Remark: In the special case of a circle (t = 3 and 2), every possible diameter can be considered a major axis. We use GeoGebra to verify our calculations. Toh Pee Choon (NIE) Session 2: Ellipse Oct / 24
10 Exercise 2.9 Use Geogebra to plot an ellipse with foci F 1 = (5, 0) and F 2 = (3, 0) and major axis of length 4. Consider an arbitrary point P on this ellipse and join the segment PF 2. By varying the position of P, deduce a relation between PF 2 and the coordinates of P. Since the foci must lie on the major axis, one of the vertex of this ellipse must lie on (6, 0). It appears that PF 2 is always half the value of the x-coordinate of P. In other words, any point P is always twice as far away from the y-axis as its distance to F 2. Define d as the line x = 0 and denote Pd as the perpendicular distance of P to d. Then Pd = 2 PF 2. To see this clearly, define D = (0, y(p)), i.e. D is on the directrix and has the y-coordinate as P. Toh Pee Choon (NIE) Session 2: Ellipse Oct / 24
11 Exercise 2.10 Let F = (0, 0) and d be the line x = 4. Find the locus of points whose distance from d is thrice the distance from F. i.e. {P : Pd = 3 PF }. (1, 0) satisfy the equation and so does ( 2, 0). It s reasonable to assume the locus is continuous. From (1, 0), where would the neighbouring points lie? Let P = (0, y) and Pd = 4, so we have y = ± 4 3. These four points are insufficient to determine a conic. (Use GeoGebra tool to see this.) For general P = (x, y), we have 3 ( x 2 + y 2 = 4 x = 8 x + 2) y 2 = 18. Toh Pee Choon (NIE) Session 2: Ellipse Oct / 24
12 Definition of Ellipse Definition 2.11 (Directrix-Eccentricity-Focus) An ellipse is the locus of points whose distance from a fixed point F (focus) and to a fixed line d (directrix) are in a constant ratio less than 1. This ratio is termed the eccentricity. i.e. { P : PF } Pd = e, 0 < e < 1. d P D F Toh Pee Choon (NIE) Session 2: Ellipse Oct / 24
13 Linking the definitions Problem 2.12 (D-E-F = Cartesian) Let F = (c, 0) and directrix d be the line x = k. Transform the equation PF Pd = e for e 1 to the form (x c ) 2 ke2 1 e 2 + y 2 1 e 2 = e2 (k c) 2 (1 e 2 ) 2. Let P = (x, y) then (x c) 2 + y 2 = e 2 (k x) 2. (Note the consequence of e = 0. Alternatively, set e = 1 k and let k.) This becomes x 2 (1 e 2 ) 2x(c ke 2 ) + y 2 = e 2 k 2 c 2. (Note the consequence of e = 1.) When 0 < e < 1, this is an ellipse thus unifying both definitions. For standard position, we set k = c/e 2. The equation simplifies to e 2 x 2 e 2 y 2 c 2 + c 2 (1 e 2 ) = 1. Toh Pee Choon (NIE) Session 2: Ellipse Oct / 24
14 Linking the definitions Definition 2.13 (D-E-F = Cartesian) Let F = (c, 0), c > 0 and directrix d be the line x = c e 2. The equation PF Pd = e for 0 < e < 1 is equivalent to e 2 x 2 e 2 y 2 c 2 + c 2 (1 e 2 ) = 1 Thus a = c e and b = (1 e 2 )a. (i.e. a > b > 0.) Directrix becomes x = a e. e = 1 b2 a 2 c = ae = a 2 b 2. (So the focus defined in the G and D-E-F definitions coincide.) By symmetry F 2 = ( c, 0) and x = a e also qualify as focus and directrix. Toh Pee Choon (NIE) Session 2: Ellipse Oct / 24
15 Linking the definitions Problem 2.14 (Cartesian = D-E-F ) Given an ellipse in standard position i.e. x2 + y 2 = 1, a b > 0 show for a 2 b 2 c = a 2 b 2 and there exist some e, satisfying 0 < e < 1 such that (x c) 2 + y 2 = e 2 ( c e 2 x ) 2. Define e = c a, which satisfies the bounds given. This gives b = a 2 c 2 = a (1 e 2 ) = c e (1 e 2 ). One can then work backwards to obtain the desired equation. In conclusion the D-E-F definition holds with F = (c, 0) and directrix x = c e 2. Toh Pee Choon (NIE) Session 2: Ellipse Oct / 24
16 PF = e Pd : Varying eccentricity and directrix Exercise 2.15 In GeoGebra, create sliders with variables 0 e 5 and 5 k 20. Create the directrix x = k and focus F = (0, 0). Create a slider 0 m 100, where m is the value of Pd. This means that P lies on one of the lines x = k + m or x = k m. (Create these two lines.) On the other hand, P lies on a circle of radius em with centre F. The intersections (four possibilities) of this circle with the previous two lines gives P. Vary m and use trace to visualize the conic. Toh Pee Choon (NIE) Session 2: Ellipse Oct / 24
17 Definition of Ellipse Definition 2.16 (Scaled Circle) Given a circle and a straight line through the centre. If we apply a scaling of factor k in the direction of the line, we obtain an ellipse. Scaling of factor b a parallel to the y-axis of the circle x 2 + y 2 = a 2. Q b P O t a P = (a cos t, b sin t) Toh Pee Choon (NIE) Session 2: Ellipse Oct / 24
18 Parametric Form Definition 2.17 In the diagram, if Q = (a cos t, a sin t) for π < t π, then the coordinates of P is P = (a cos t, b sin t). Important: t is not equal to θ, the angle between OP and the x-axis. Q b P O t θ a tan θ = b a tan t Toh Pee Choon (NIE) Session 2: Ellipse Oct / 24
19 Linking the definitions Problem 2.18 (Cartesian = Geometric ) Let x2 = 1 be an ellipse with a b > 0 and let F b 2 1 be the right focus. Show that for an arbitrary point P = (a cos t, b sin t), a 2 + y 2 PF 1 = a(1 e cos t). PF 1 = (a cos t a 2 b 2 ) 2 + b 2 sin 2 t = a 2 cos 2 t 2a(cos t) a 2 b 2 + a 2 b 2 + b 2 (1 cos 2 t) ( a = 2 b 2 cos t a ( 2 = a 1 b2 cos t 1) = a(1 e cos t). a 2 ) 2 Similarly PF 2 = a(1 + e cos t). Consequently, PF 1 + PF 2 = 2a. Toh Pee Choon (NIE) Session 2: Ellipse Oct / 24
20 Reflection Property Problem 2.19 Given an ellipse in standard position with foci F 1 and F 2. Let P = (x 0, y 0 ) be a point on the ellipse and N be on the x-axis such that PN is normal to the tangent line of the ellipse at P. Show that F 2 PN = NPF 1. P N F 2 F1 Aim: Show PF 2 NF 2 = PF 1 NF 1. Toh Pee Choon (NIE) Session 2: Ellipse Oct / 24
21 Proof of Reflection Property Theorem 2.20 (Angle Bisector Theorem) In ABC with D on BC,AD bisects BAC if and only if AB BD = AC CD. Aim: Show PF 2 NF 2 = PF 1 NF 1. We have PF 1 = a ex 0 and PF 2 = a + ex 0. Differentiating implicitly, we have 2 x a y b 2 dy dx = 0 = dy dx = xb2 ya 2. So the gradient of PN is given by a2 y 0 b 2 x 0. Let N = (x 1, 0), then ) y 0 = a2 y 0 x 0 x 1 b 2 = x 1 = x 0 (1 b2 x 0 a 2 = e 2 x 0. NF 1 = c x 1 = ae e 2 x 0 = e(a ex 0 ) and NF 2 = x 1 + c = e 2 x 0 + ea = e(a + ex 0 ). Toh Pee Choon (NIE) Session 2: Ellipse Oct / 24
22 Polar form with respect to the focus Problem 2.21 Let F 1 = (0, 0) be a focus and x = k, k > 0 be the directrix. Find the polar form of the resulting ellipse with eccentricity e. r P F2 F1 θ k PF 1 = r and P d = k r cos θ. So r = e(k r cos θ) = r = ek 1 + e cos θ. Toh Pee Choon (NIE) Session 2: Ellipse Oct / 24
23 Problem 2.22 In the diagram AB is the major axis of an ellipse in standard position. The tangent to the ellipse at P intersects the tangents to vertices A and B at C and D respectively. Show that PF 1 PF 2 = PC PD. C P D A F 2 F 1 B Toh Pee Choon (NIE) Session 2: Ellipse Oct / 24
24 Problem 2.23 Cut out a paper disc and mark out an arbitrary interior point X distinct from the centre of the disc. Now pick a random point P on the circumference of the disc and and fold the disc (with one crease) such that P touches X. Use a pencil to mark out the crease. Repeat for as many points P as possible. What do you observe? Explain. Toh Pee Choon (NIE) Session 2: Ellipse Oct / 24
Conic 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 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 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 informationAdditional Mathematics Lines and circles
Additional Mathematics Lines and circles Topic assessment 1 The points A and B have coordinates ( ) and (4 respectively. Calculate (i) The gradient of the line AB [1] The length of the line AB [] (iii)
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 informationby Abhijit Kumar Jha
SET I. If the locus of the point of intersection of perpendicular tangents to the ellipse x a circle with centre at (0, 0), then the radius of the circle would e a + a /a ( a ). There are exactl two points
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 informationIIT JEE Maths Paper 2
IIT JEE - 009 Maths Paper A. Question paper format: 1. The question paper consists of 4 sections.. Section I contains 4 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for
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 informationTopic 2 [312 marks] The rectangle ABCD is inscribed in a circle. Sides [AD] and [AB] have lengths
Topic 2 [312 marks] 1 The rectangle ABCD is inscribed in a circle Sides [AD] and [AB] have lengths [12 marks] 3 cm and (\9\) cm respectively E is a point on side [AB] such that AE is 3 cm Side [DE] is
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 informationObjective Mathematics
. A tangent to the ellipse is intersected by a b the tangents at the etremities of the major ais at 'P' and 'Q' circle on PQ as diameter always passes through : (a) one fied point two fied points (c) four
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 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 information5 Find an equation of the circle in which AB is a diameter in each case. a A (1, 2) B (3, 2) b A ( 7, 2) B (1, 8) c A (1, 1) B (4, 0)
C2 CRDINATE GEMETRY Worksheet A 1 Write down an equation of the circle with the given centre and radius in each case. a centre (0, 0) radius 5 b centre (1, 3) radius 2 c centre (4, 6) radius 1 1 d centre
More informationLecture 17. Implicit differentiation. Making y the subject: If xy =1,y= x 1 & dy. changed to the subject of y. Note: Example 1.
Implicit differentiation. Lecture 17 Making y the subject: If xy 1,y x 1 & dy dx x 2. But xy y 2 1 is harder to be changed to the subject of y. Note: d dx (f(y)) f (y) dy dx Example 1. Find dy dx given
More information1 is equal to. 1 (B) a. (C) a (B) (D) 4. (C) P lies inside both C & E (D) P lies inside C but outside E. (B) 1 (D) 1
Single Correct Q. Two mutuall perpendicular tangents of the parabola = a meet the ais in P and P. If S is the focus of the parabola then l a (SP ) is equal to (SP ) l (B) a (C) a Q. ABCD and EFGC are squares
More informationl (D) 36 (C) 9 x + a sin at which the tangent is parallel to x-axis lie on
Dpp- to MATHEMATICS Dail Practice Problems Target IIT JEE 00 CLASS : XIII (VXYZ) DPP. NO.- to DPP- Q. If on a given base, a triangle be described such that the sum of the tangents of the base angles is
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 informationHonors Precalculus Chapter 8 Summary Conic Sections- Parabola
Honors Precalculus Chapter 8 Summary Conic Sections- Parabola Definition: Focal length: y- axis P(x, y) Focal chord: focus Vertex x-axis directrix Focal width/ Latus Rectum: Derivation of equation of parabola:
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 information+ 2gx + 2fy + c = 0 if S
CIRCLE DEFINITIONS A circle is the locus of a point which moves in such a way that its distance from a fixed point, called the centre, is always a constant. The distance r from the centre is called the
More informationAnalytic 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 informationConic Sections: THE ELLIPSE
Conic Sections: THE ELLIPSE An ellipse is the set of all points,such that the sum of the distance between, and two distinct points is a constant. These two distinct points are called the foci (plural of
More informationCURVATURE AND RADIUS OF CURVATURE
CHAPTER 5 CURVATURE AND RADIUS OF CURVATURE 5.1 Introduction: Curvature is a numerical measure of bending of the curve. At a particular point on the curve, a tangent can be drawn. Let this line makes an
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 informationAnalytic 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 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 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 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 informationFrom the SelectedWorks of Harish Chandra Rajpoot H.C. Rajpoot. Harish Chandra Rajpoot Rajpoot, HCR. Winter February 24, 2015
From the SelectedWorks of Harish Chandra Rajpoot H.C. Rajpoot Winter February 24, 2015 Mathematical Analysis of Elliptical Path in the Annular Region Between Two Circles, Smaller Inside the Bigger One
More informationPractice Assessment Task SET 3
PRACTICE ASSESSMENT TASK 3 655 Practice Assessment Task SET 3 Solve m - 5m + 6 $ 0 0 Find the locus of point P that moves so that it is equidistant from the points A^-3, h and B ^57, h 3 Write x = 4t,
More informationRecognise the Equation of a Circle. Solve Problems about Circles Centred at O. Co-Ordinate Geometry of the Circle - Outcomes
1 Co-Ordinate Geometry of the Circle - Outcomes Recognise the equation of a circle. Solve problems about circles centred at the origin. Solve problems about circles not centred at the origin. Determine
More informationCircles, Mixed Exercise 6
Circles, Mixed Exercise 6 a QR is the diameter of the circle so the centre, C, is the midpoint of QR ( 5) 0 Midpoint = +, + = (, 6) C(, 6) b Radius = of diameter = of QR = of ( x x ) + ( y y ) = of ( 5
More informationx n+1 = ( x n + ) converges, then it converges to α. [2]
1 A Level - Mathematics P 3 ITERATION ( With references and answers) [ Numerical Solution of Equation] Q1. The equation x 3 - x 2 6 = 0 has one real root, denoted by α. i) Find by calculation the pair
More informationUNIT 6. BELL WORK: Draw 3 different sized circles, 1 must be at LEAST 15cm across! Cut out each circle. The Circle
UNIT 6 BELL WORK: Draw 3 different sized circles, 1 must be at LEAST 15cm across! Cut out each circle The Circle 1 Questions How are perimeter and area related? How are the areas of polygons and circles
More informationEdexcel GCE A Level Maths. Further Maths 3 Coordinate Systems
Edecel GCE A Level Maths Further Maths 3 Coordinate Sstems Edited b: K V Kumaran kumarmaths.weebl.com 1 kumarmaths.weebl.com kumarmaths.weebl.com 3 kumarmaths.weebl.com 4 kumarmaths.weebl.com 5 1. An ellipse
More informationJEE-ADVANCED MATHEMATICS. Paper-1. SECTION 1: (One or More Options Correct Type)
JEE-ADVANCED MATHEMATICS Paper- SECTION : (One or More Options Correct Type) This section contains 8 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE OR
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 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 informationEdexcel New GCE A Level Maths workbook Circle.
Edexcel New GCE A Level Maths workbook Circle. Edited by: K V Kumaran kumarmaths.weebly.com 1 Finding the Midpoint of a Line To work out the midpoint of line we need to find the halfway point Midpoint
More informationRMT 2014 Geometry Test Solutions February 15, 2014
RMT 014 Geometry Test Solutions February 15, 014 1. The coordinates of three vertices of a parallelogram are A(1, 1), B(, 4), and C( 5, 1). Compute the area of the parallelogram. Answer: 18 Solution: Note
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 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 informationMATHEMATICS 2017 HSC Course Assessment Task 4 (Trial Examination) Thursday, 3 August 2017
MATHEMATICS 2017 HSC Course Assessment Task 4 (Trial Examination) Thursday, 3 August 2017 General instructions Working time 3 hours. (plus 5 minutes reading time) Write using blue or black pen. Where diagrams
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 informationCircle geometry investigation: Student worksheet
Circle geometry investigation: Student worksheet http://topdrawer.aamt.edu.au/geometric-reasoning/good-teaching/exploringcircles/explore-predict-confirm/circle-geometry-investigations About these activities
More informationSM2H Unit 6 Circle Notes
Name: Period: SM2H Unit 6 Circle Notes 6.1 Circle Vocabulary, Arc and Angle Measures Circle: All points in a plane that are the same distance from a given point, called the center of the circle. Chord:
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 information( 1 ) Show that P ( a, b + c ), Q ( b, c + a ) and R ( c, a + b ) are collinear.
Problems 01 - POINT Page 1 ( 1 ) Show that P ( a, b + c ), Q ( b, c + a ) and R ( c, a + b ) are collinear. ( ) Prove that the two lines joining the mid-points of the pairs of opposite sides and the line
More informationJUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 10 (Second moments of an arc) A.J.Hobson
JUST THE MATHS UNIT NUMBER 13.1 INTEGRATION APPLICATIONS 1 (Second moments of an arc) by A.J.Hobson 13.1.1 Introduction 13.1. The second moment of an arc about the y-axis 13.1.3 The second moment of an
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 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 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 information3.4 Conic sections. Such type of curves are called conics, because they arise from different slices through a cone
3.4 Conic sections Next we consider the objects resulting from ax 2 + bxy + cy 2 + + ey + f = 0. Such type of curves are called conics, because they arise from different slices through a cone Circles belong
More informationChapter 10. Properties of Circles
Chapter 10 Properties of Circles 10.1 Use Properties of Tangents Objective: Use properties of a tangent to a circle. Essential Question: how can you verify that a segment is tangent to a circle? Terminology:
More informationClass IX : Math Chapter 11: Geometric Constructions Top Concepts 1. To construct an angle equal to a given angle. Given : Any POQ and a point A.
1 Class IX : Math Chapter 11: Geometric Constructions Top Concepts 1. To construct an angle equal to a given angle. Given : Any POQ and a point A. Required : To construct an angle at A equal to POQ. 1.
More informationMATHEMATICS Code No. 13 INSTRUCTIONS
DO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE ASKED TO DO SO COMBINED COMPETITIVE (PRELIMINARY) EXAMINATION, 00 Serial No. MATHEMATICS Code No. A Time Allowed : Two Hours Maximum Marks : 00 INSTRUCTIONS.
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 informationChapter (Circle) * Circle - circle is locus of such points which are at equidistant from a fixed point in
Chapter - 10 (Circle) Key Concept * Circle - circle is locus of such points which are at equidistant from a fixed point in a plane. * Concentric circle - Circle having same centre called concentric circle.
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 informationCore Mathematics 2 Coordinate Geometry
Core Mathematics 2 Coordinate Geometry Edited by: K V Kumaran Email: kvkumaran@gmail.com Core Mathematics 2 Coordinate Geometry 1 Coordinate geometry in the (x, y) plane Coordinate geometry of the circle
More informationCO-ORDINATE GEOMETRY
CO-ORDINATE GEOMETRY 1 To change from Cartesian coordinates to polar coordinates, for X write r cos θ and for y write r sin θ. 2 To change from polar coordinates to cartesian coordinates, for r 2 write
More informationMATH10000 Mathematical Workshop Project 2 Part 1 Conic Sections
MATH10000 Mathematical Workshop Project 2 Part 1 Conic Sections The aim of this project is to introduce you to an area of geometry known as the theory of conic sections, which is one of the most famous
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 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 informationCONCURRENT LINES- PROPERTIES RELATED TO A TRIANGLE THEOREM The medians of a triangle are concurrent. Proof: Let A(x 1, y 1 ), B(x, y ), C(x 3, y 3 ) be the vertices of the triangle A(x 1, y 1 ) F E B(x,
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 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 informationPRACTICE PAPER 6 SOLUTIONS
PRACTICE PAPER 6 SOLUTIONS SECTION A I.. Find the value of k if the points (, ) and (k, 3) are conjugate points with respect to the circle + y 5 + 8y + 6. Sol. Equation of the circle is + y 5 + 8y + 6
More informationTrans Web Educational Services Pvt. Ltd B 147,1st Floor, Sec-6, NOIDA, UP
Solved Examples Example 1: Find the equation of the circle circumscribing the triangle formed by the lines x + y = 6, 2x + y = 4, x + 2y = 5. Method 1. Consider the equation (x + y 6) (2x + y 4) + λ 1
More informationIntermediate Math Circles Wednesday, April 5, 2017 Problem Set 8
Intermediate Math Circles Wednesday, April 5, 2017 Problem Set 8 1. Determine the coordinates of the vertices and foci for each of the following ellipses. (a) + 9y 2 = 36 We want equation to be of the
More informationa Write down the coordinates of the point on the curve where t = 2. b Find the value of t at the point on the curve with coordinates ( 5 4, 8).
Worksheet A 1 A curve is given by the parametric equations x = t + 1, y = 4 t. a Write down the coordinates of the point on the curve where t =. b Find the value of t at the point on the curve with coordinates
More information10. Circles. Q 5 O is the centre of a circle of radius 5 cm. OP AB and OQ CD, AB CD, AB = 6 cm and CD = 8 cm. Determine PQ. Marks (2) Marks (2)
10. Circles Q 1 True or False: It is possible to draw two circles passing through three given non-collinear points. Mark (1) Q 2 State the following statement as true or false. Give reasons also.the perpendicular
More informationSenior Math Circles February 11, 2009 Conics II
1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Senior Math Circles February 11, 2009 Conics II Locus Problems The word locus is sometimes synonymous with
More informationAS PURE MATHS REVISION NOTES
AS PURE MATHS REVISION NOTES 1 SURDS A root such as 3 that cannot be written exactly as a fraction is IRRATIONAL An expression that involves irrational roots is in SURD FORM e.g. 2 3 3 + 2 and 3-2 are
More information[VECTOR AND COORDINATE GEOMETRY, EDO UNIVERSITY, IYAMHO] P a g e 1
COURSE CODE: MTH 112 COURSE TITLE: VECTOR AND COORDINATE GEOMETRY NUMBER OF UNITS: 3 Units COURSE DURATION: Three hours per week COURSE LECTURER: ALHASSAN CHARITY INTENDED LEARNING OUTCOMES At the end
More informationDevice Constructions with Hyperbolas
lfonso Croeze 1 William Kelly 1 William Smith 2 1 Department of Mathematics Louisiana State University aton Rouge, L 2 Department of Mathematics University of Mississippi Oxford, MS July 8, 2011 Hyperbola
More information2. A die is rolled 3 times, the probability of getting a number larger than the previous number each time is
. If P(A) = x, P = 2x, P(A B) = 2, P ( A B) = 2 3, then the value of x is (A) 5 8 5 36 6 36 36 2. A die is rolled 3 times, the probability of getting a number larger than the previous number each time
More informationENGI Parametric Vector Functions Page 5-01
ENGI 3425 5. Parametric Vector Functions Page 5-01 5. Parametric Vector Functions Contents: 5.1 Arc Length (Cartesian parametric and plane polar) 5.2 Surfaces of Revolution 5.3 Area under a Parametric
More informationDEPARTMENT OF MATHEMATICS
DEPARTMENT OF MATHEMATICS AS level Mathematics Core mathematics 2 - C2 2015-2016 Name: Page C2 workbook contents Algebra Differentiation Integration Coordinate Geometry Logarithms Geometric series Series
More information698 Chapter 11 Parametric Equations and Polar Coordinates
698 Chapter Parametric Equations and Polar Coordinates 67. 68. 69. 70. 7. 7. 7. 7. Chapter Practice Eercises 699 75. (a Perihelion a ae a( e, Aphelion ea a a( e ( Planet Perihelion Aphelion Mercur 0.075
More informationQuestion. [The volume of a cone of radius r and height h is 1 3 πr2 h and the curved surface area is πrl where l is the slant height of the cone.
Q1 An experiment is conducted using the conical filter which is held with its axis vertical as shown. The filter has a radius of 10cm and semi-vertical angle 30. Chemical solution flows from the filter
More informationMath : Analytic Geometry
7 EP-Program - Strisuksa School - Roi-et Math : Analytic Geometry Dr.Wattana Toutip - Department of Mathematics Khon Kaen University 00 :Wattana Toutip wattou@kku.ac.th http://home.kku.ac.th/wattou 7 Analytic
More informationMath & 8.7 Circle Properties 8.6 #1 AND #2 TANGENTS AND CHORDS
Math 9 8.6 & 8.7 Circle Properties 8.6 #1 AND #2 TANGENTS AND CHORDS Property #1 Tangent Line A line that touches a circle only once is called a line. Tangent lines always meet the radius of a circle at
More informationCircles MODULE - II Coordinate Geometry CIRCLES. Notice the path in which the tip of the hand of a watch moves. (see Fig. 11.1)
CIRCLES Notice the path in which the tip of the hand of a watch moves. (see Fig..) 0 9 3 8 4 7 6 5 Fig.. Fig.. Again, notice the curve traced out when a nail is fied at a point and a thread of certain
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 informationPossible C2 questions from past papers P1 P3
Possible C2 questions from past papers P1 P3 Source of the original question is given in brackets, e.g. [P1 January 2001 Question 1]; a question which has been edited is indicated with an asterisk, e.g.
More informationP1 Chapter 6 :: Circles
P1 Chapter 6 :: Circles jfrost@tiffin.kingston.sch.uk www.drfrostmaths.com @DrFrostMaths Last modified: 11 th August 2017 Use of DrFrostMaths for practice Register for free at: www.drfrostmaths.com/homework
More informationMATHEMATICS EXTENSION 2
PETRUS KY COLLEGE NEW SOUTH WALES in partnership with VIETNAMESE COMMUNITY IN AUSTRALIA NSW CHAPTER JULY 006 MATHEMATICS EXTENSION PRE-TRIAL TEST HIGHER SCHOOL CERTIFICATE (HSC) Student Number: Student
More informationUSA Mathematics Talent Search
16 3 1 (a) Since x and y are 3-digit integers, we begin by noting that the condition 6(x y) = (y x) is equivalent to 6(1, 000x + y) = 1, 000y + x = 5, 999x = 994y, which (after factoring out a 7 by long
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 informationHanoi Open Mathematical Competition 2017
Hanoi Open Mathematical Competition 2017 Junior Section Saturday, 4 March 2017 08h30-11h30 Important: Answer to all 15 questions. Write your answers on the answer sheets provided. For the multiple choice
More informationSec 4 Maths SET D PAPER 2
S4MA Set D Paper Sec 4 Maths Exam papers with worked solutions SET D PAPER Compiled by THE MATHS CAFE P a g e Answer all questions. Write your answers and working on the separate Answer Paper provided.
More informationUdaan School Of Mathematics Class X Chapter 10 Circles Maths
Exercise 10.1 1. Fill in the blanks (i) The common point of tangent and the circle is called point of contact. (ii) A circle may have two parallel tangents. (iii) A tangent to a circle intersects it in
More informationCreated by T. Madas LINE INTEGRALS. Created by T. Madas
LINE INTEGRALS LINE INTEGRALS IN 2 DIMENSIONAL CARTESIAN COORDINATES Question 1 Evaluate the integral ( x + 2y) dx, C where C is the path along the curve with equation y 2 = x + 1, from ( ) 0,1 to ( )
More informationMath 234. What you should know on day one. August 28, You should be able to use general principles like. x = cos t, y = sin t, 0 t π.
Math 234 What you should know on day one August 28, 2001 1 You should be able to use general principles like Length = ds, Area = da, Volume = dv For example the length of the semi circle x = cos t, y =
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 information1 k. cos tan? Higher Maths Non Calculator Practice Practice Paper A. 1. A sequence is defined by the recurrence relation u 2u 1, u 3.
Higher Maths Non Calculator Practice Practice Paper A. A sequence is defined b the recurrence relation u u, u. n n What is the value of u?. The line with equation k 9 is parallel to the line with gradient
More informationSimple Co-ordinate geometry problems
Simple Co-ordinate geometry problems 1. Find the equation of straight line passing through the point P(5,2) with equal intercepts. 1. Method 1 Let the equation of straight line be + =1, a,b 0 (a) If a=b
More information