Chapter 1 Analytic geometry in the plane

Transcription

1 3110 General Mathematics 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: The study of motion has been important since ancient times. Calculus is the mathematical tool to describe it. Conic sections are the paths traveled by planets, satellites and other bodies (even electrons). Figure 1 (Conic sections) Topics: 1.1 Conic Sections 1. Translation of axis 1.3 Rotation of axis

2 3110 General Mathematics 1.1 Conic Sections Circles Definitions A circle is the set of points in a plane whose distance from a given fixed point in the plane is constant. The fixed point is the center of the circle; the constant distance is the radius. How to draw a circle. - Compass - String How to find the equation for a circle. The standard-form equation for the circle of radius a centered at the origin is x y a

3 3110 General Mathematics 3 Example 1 Find the center and radius of the circle x y 9. Then sketch the circle. Include the center and radius in the sketch. Solution Parabolas Definitions A parabola is the set that consists of all the points in a plane equidistant from a given fixed point and a given fixed line in the plane. The fixed point is the focus of the parabola. The fixed line is the directrix. How to draw a parabola.

4 3110 General Mathematics 4 How to find the equation for a parabola. The standard-form equations for parabolas with vertices at the origin and p 0 are shown in the table below. Equation Focus Directrix Axis Opens x 4 py (0, p ) y p y-axis Up x 4 py (0, p) y p y-axis Down y 4 px ( p,0) x p x-axis To the right y 4 px ( p,0) x p x-axis To the left

5 3110 General Mathematics 5 Example Find the focus and directrix of the parabola y 10x. Then sketch the parabola. Include the focus and directrix in the sketch. Solution Ellipses Definitions An ellipse is the set of points in a plane whose distances from two fixed points in the plane have a constant sum. The two fixed points are the foci of the ellipse. How to draw an ellipse.

6 3110 General Mathematics 6 How to find the equation for an ellipse. The standard-form equations for ellipses centered at the origin with a b 0 and c a b Equation Foci Vertices Major axis x a x b y b c,0 a,0 1 y a 0, c 0, a 1 x-axis y-axis

7 3110 General Mathematics 7 x y Example 3 Find the foci and vertices of the ellipse 1. Then 16 9 sketch the ellipse. Include the foci and vertices in the sketch. Solution Remark: Consider the ellipse x a y, with c a b b 1 1. If c 0 (so that a b) then the ellipse will be a circle.. If c a (so that b 0) then the ellipse will be a line segment Hyperbolas Definitions A hyperbola is the set of points in a plane whose distances from two fixed points in the plane have a constant difference. The two fixed points are the foci of the hyperbola. How to draw a hyperbola.

8 3110 General Mathematics 8 How to find the equation for a hyperbola. How to graph a hyperbola. The standard-form equations for hyperbolas centered at the origin with a 0, b 0 and c a b Equation Foci Vertices Asymptote x a y b c,0 a,0 1 y b a x y a x b 0, c 0, a 1 a y b x

9 3110 General Mathematics 9 Example 4 Find the foci and asymptotes of the hyperbola x y Then sketch the hyperbola. Include the foci, vertices and asymptotes in the sketch. Solution

10 3110 General Mathematics 10 How to classify conic sections by Eccentricity. Eccentricity Definition An eccentricity of a conic section is the constant ratio of the distance between the conic section and the focus to the distance between the conic section and the directrix. How to classify conic sections by Eccentricity. Theorem If e is the eccentricity of a conic section, then the conic section is: (a) parabola if e 1 (b) ellipse if e 1 (c) hyperbola if e 1 Outline proof (a)

11 3110 General Mathematics 11 Outline proof (b)

12 3110 General Mathematics 1 Outline proof (c) Remark: In both ellipse and hyperbola, the eccentricity is the ratio of the distance between the foci and to the distance between the vertices. c Eccentricity a

13 3110 General Mathematics 13 Example 5 Find the equation and sketch the graph for an ellipse of the 1 eccentricity e whose foci lie at the points (1,0) and ( 1,0). Example 6 Find the equation and sketch the graph for a hyperbola of 5 eccentricity e whose vertices locate at the points (0,3)and (0, 3). 3

14 3110 General Mathematics Translation of axis How to translate ( or shift) a graph of y f ( x) Example 1..1 Sketch the graphs of the following equations: y x y x y 1 x y ( x 1) y ( x )

15 3110 General Mathematics 15 Rules for translating of axis On the system of rectangular coordinate XY with the origin O, we can construct a new system X Y with the origin O as in the figure. Y Y (h, k) O h O k X X The Relations between ( xy, ) and ( x, y ) are the following: or equivalently, x x h y y k x x h y y k (1.1) (1.) Applying equation (1.) we obtain some conclusions as the following: - To shift the graph of y f ( x) straight up k unit, we add k to y - To shift the graph of y f ( x) down k unit, we add k to y - To shift the graph of y f ( x) to the right k unit, we add k to x - To shift the graph of y f ( x) to the left k unit, we add k to x Example 1.. Change the equation y x straight up 4 units. The sketch then graph. in order to shift its graph

16 3110 General Mathematics 16 Example 1..3 Change the equation x y 4 in order to shift its graph down 3 units. Then sketch the graph. Consequently, we obtain the standard-form equations for conic sections as the followings: The standard-form equation for the circle of radius a centered at the point ( hk, ) is ( x h) ( y k) a The standard-form equations for parabolas with vertices at the point ( hk, ) and p 0 are shown in the table below. Equation Focus Directrix Axis ( x h) 4 p( y k) ( h, k p) y k p ( x h) 4 p( y k) x h ( h, k p) y k p x h ( h p, k) x h p y k ( y k) 4 p( x h) ( h p, k) x h p y k ( y k) 4 p( x h)

17 3110 General Mathematics 17 The standard-form equations for ellipses centered at the point ( hk, ) with a b 0 and c a b Equation Foci Vertices Major axis ( x h) ( y k) 1 h c, k a b h a, k y k ( x h) ( y k) 1 h, k c b a h, k a x h The standard-form equations for hyperbolas centered at the point ( hk, ) with a 0, b 0 and c a b Equation Foci Vertices ( x h) ( y k) 1 h c, k a b ( y k) ( x h) 1 h, k c a b h a, k h, k a

18 3110 General Mathematics 18 Example 1..4 Find the center and radius of the circle x y 4x 6y 3 0. Then sketch the graph. Example 1..5 Find the focus and the vertex of the parabola x 6x 8y 5 0. Then sketch the graph.

19 3110 General Mathematics 19 Example 1..6 Find the standard form of the conic section 4x 9y 8x 36y 4 0. Then sketch the graph. Example 1..7 Find the standard form of the conic section 9x 4y 18x 16y 9 0. Then sketch the graph.

20 3110 General Mathematics 0 Quadratic Equations A general form of quadratic equation may be written as Ax Bxy Cy Dx Ey F 0 (1.3) in which A, B and C are not all zero. In this section we have seen that if the axis of a conic section parallel to the coordinate axis then the equation of the conic section is in the form Ax Cy Dx Ey F 0 (1.4) in which the cross product term, Bxy, did not appear. We can apply completing the squares to identify the equation. We may have noticed that the graph of equation (1.4) is a (or an) a) parabola if AC 0 b) ellipse if AC 0 c) hyperbola if AC 0 Let s consider the graph of the hyperbola xy 9 Note that, the graph is rotated through an angle of 4 radians from the x- axis about the origin. In the next section we will discuss on rotating of axes.

21 3110 General Mathematics Rotation of axes Let a new coordinate XY be a counterclockwise rotation through angle about the origin of the coordinate XY as in the figure. The relations between (, ) xy and ( x, y ) are as follow: x x cos y sin (1.5) y x sin y cos (1.6)

22 3110 General Mathematics Example The x- and y-axes are rotated through an angle of 4 radians about the origin. Find an equation for the hyperbola 9 xy in the new coordinates.

23 3110 General Mathematics 3 Notice that the equation of rotation can be solved to obtain the inverse relation as How come? x xcos ysin (1.7) y y cos xsin (1.8)

24 3110 General Mathematics 4 Example 1.3. Find an equation for the ellipse whose foci located at the point ( 6, 6) and ( 6, 6) with the constant sum 14 in the original coordinate.

25 3110 General Mathematics 5 Notice that if we apply the equation of rotation in the general quadratic equation Ax Bxy Cy Dx Ey F 0 (1.9) then we obtain a new quadratic equation in the coordinate XY as A x B x y C y D x E y F 0 (1.10) The new and old coefficients are related by the equations Why? A Acos Bcos sin C sin B Bcos ( C A)sin C Asin Bcos sin C cos D Dcos E sin E Dsin E cos F F

26 3110 General Mathematics 6 Consequently, we obtain a wonderful theorem for moving the cross product term from the new quadratic equation as the following: Theorem Given Ax Bxy Cy Dx Ey F 0 be a quadratic equation in the coordinate XY and B 0. If the coordinate A C XY is rotated through an angle and cot then the equation B in the new coordinate is reduced in the form A x C y D x E y F 0 in which the cross product term did not appear. Proof

27 3110 General Mathematics 7 Example The coordinate axes are to be rotated through an angle to produce an equation for the curve x 3xy y 10 0 that has no cross product term. Find and the new equation. Identify the curve.

28 3110 General Mathematics 8 Example Identify the graph of the equation x 3xy y 5 0

29 3110 General Mathematics 9 Since axes can always be rotated to eliminate the cross product term, then we might be consider any quadratic equations in the form Ax Cy Dx Ey F 0 This form represents a) a circle if A C 0 (special cases: a point or no graph) b) a parabola if A 0 and C 0 or otherwise A 0 and C 0 c) an ellipse if A and C are both positive or both negative (special cases: a circle, a single point or no graph at all) d) a hyperbola if A and C have opposite signs (special case: a pair of intersection lines) e) a straight line if A C 0 and at least D and E is different from zero f) one or two straight line if the left hand side of the equation can be factored into the product of two linear factors However, we do not need to eliminate the xy-term from the equation Ax Bxy Cy Dx Ey F 0 to tell what kind of conic section the equation represents. We can use the discriminant test as stated in the following theorem:

30 3110 General Mathematics 30 Theorem 1.3. Let B 4AC be the discriminant of a quadratic equation Ax Bxy Cy Dx Ey F 0 Then the curve of the equation is a) a parabola if B 4AC 0 b) an ellipse if B 4AC 0 c) a hyperbola if B 4AC 0 Proof Example Fill in the blanks (a) (b) (c) 3x 6xy 3y x 7 0 represents a because. x xy y 1 0 represents a because. xy y 5y 1 0 represents a because.

31 3110 General Mathematics 31 Chapter Vectors.1 Review of vectors. Linear combination and linearly independent.3 Vectors in three dimensional space Chapter 3 Limit and continuity of functions Chapter 4 Derivative of functions Chapter 5 Applications of derivative and differentials Chapter 6 Integration Chapter 7 Applications of integration Chapter 8 Differential equations