1. WARM UP 1.Find the equation of the circle with center at (9, 2) and radius 2. 2.Find the center and radius of the circle 3.Find the center and radius of the circle. (10, -3) 10 (-6, -2) 7

2. ELIPSES

3. OBJECTIVES Find the vertices and foci, and draw a graph of an ellipse, given its equation. Use the technique of completing the square to find the center, vertices, and foci and draw a graph of an ellipse. Solve problems involving ellipses

4. ELLIPSES An interesting attraction found in museums is the whispering gallery. It is elliptical. Persons with their head at the foci can whisper and hear each other clearly, while persons at other positions cannot hear them. This happens because sound waves emanating from one focus are all reflected to the other focus. Whispering gallery

5. EQUATIONS OF ELLIPSES Some equations of second degree have graphs that are ellipses. Definition An ellipse is the set of all points P in a plane such that the sum of the distances from P to two fixed points and is constant. Each fixed point is called a focus (plural: foci) of the ellipse, and shown in the diagram at the right are called focal radii.

6. REAL WORLD APPLICATIONS Ellipses have many other applications. Earth satellites travel around Earth in elliptical orbits. The planets of the solar system travel around the sun in elliptical orbits with the sun located at one focus. Jupiter, the 5th planet from the sun. The orbits of Earth and Jupiter are nearly circular. In millions of kilometers, Earth’s mean distance from the sun is 149.6, while Jupiter’s is 778.3. About how much longer is Jupiter’s orbit than Earth’s?

7. AXES OF SYMMETRY An ellipse has two axes of symmetry. The longer axis is the major axis, and the shorter axis is the minor axis. The axes are perpendicular at their midpoints, the center of the ellipse. Major axis Minor axis cente r Vertices Foci The foci lie on the major axis. The ellipse intersects the major and minor axes at the vertices.

8. EQUATION OF AN ELLIPSE We first consider an equation of an ellipse whose center is at the origin with foci on either the x- or y-axis. For foci on the x-axis the ellipse is horizontal. Major axis (0, b) (c, 0) (a, 0) (-c, 0) (-a, 0) (0, -b) For foci on the y-axis the ellipse is vertical. (0, a) (0, -a) (b, 0) (-b, 0) Major axis (0, c) (0, -c) y x

9. EQUATION OF AN ELLIPSE The sum of the focal radii is constant and is equal to the length of the major axis. (c, 0) (a, 0) (-c, 0) (-a, 0) P In the graph at the right Theorem 10-5 The equation, in standard form, of an ellipse centered at the origin with foci on an axis and c units from the origin is (major axis horizontal), or (major axis vertical) where vertical) where

10. EXAMPLE 1 For the ellipse x + 16y = 16, find the vertices and foci and draw a graph. We first multiply by 1/16 to find standard form. Since the denominator for x is larger, the foci are on the x-axis, and the ellipse is horizontal. Thus a = 4, b = 1. The vertices on the major axis are (-4, 0) and (4, 0). The other vertices are (0, -1) and (0, 1). or

11. EXAMPLE 1 CONTINUED c = a – b, so c = 16 – 1, c = √15 and the foci are (√15, 0) and (-√15, 0). To graph, we plot the vertices and draw a smooth curve. (-4, 0) (4, 0) (0, 1) (0, -1)(-√15, 0)(√15, 0)

12. TRY THIS… For each ellipse find the vertices and foci, and draw a graph. 1.x + 9y = 9 2.9x + 25y = 225 3.2x + 4y = 8

13. EXAMPLE 2 For the ellipse 9x + 2y = 18, find the vertices and foci and draw a graph. a) We first multiply by 1/18 to find standard form. b) Since the denominator for y is larger, the foci are on the y-axis and the ellipse is vertical. Thus a = 3 and b = √2. The vertices are on the major axis are (0, -3) and (0, 3). The other vertices are (-√2, 0) and (√2, 0). c = a – b, so c = 9 – 2, c = √7, and the foci are (0, -√7) and (0, √7). or

14. EXAMPLE 1 CONTINUED The graph of the equation is. (-√2, 0) (√2, 0) (0, 3) (0, -3) (0,-√7) (0,√7)

15. TRY THIS… For each ellipse find the center, vertices and foci, and draw a graph. 1.9x + y = 9 2.25x + 9y = 225 3.4x + 2y = 8 4.x + 3y = 48

16. COMPLETING THE SQUARE TO FIND STANDARD FORM If the center of an ellipse is not at the origin but at some point (h, k), then the standard form of the equation is Theorem 10-6 The equation in standard form, of an ellipse centered at (h, k) with foci c units from (h, k) is (major axis horizontal) or (major axis horizontal) or (major axis vertical) (major axis vertical) where c = a – b

17. EXAMPLE 3 For the ellipse 16x + 4y + 96x – 8y + 84= 0, find the center, vertices and foci and draw a graph. a) We first complete the square to get standard form. 16(x + 6x ) + 4(y – 2y ) = -84 16(x + 6x + 9) + 4(y – 2y + 1) = -84 + 16  9 + 4  1 16(x + 6x + 9) + 4(y – 2y + 1) = -84 + 144 + 4 16(x +3) + 4(y – 1) = 64 Multiplying 1/64 to make the right side 1. Using the addition property The center is (-3, 1), a = 4 and b = 2. The major axis is vertical.

18. EXAMPLE 3 CONTINUED b) The vertices of are (2, 0), (-2, 0), (0, 4) and (0, -4). Since, and its foci are (0, 2√3) and (0, - 2√3). Since, and its foci are (0, 2√3) and (0, - 2√3). This ellipse is centered at the origin. Thus, we need to translate the ellipse so that the center is at (-3, 1). c) The vertices and foci of the translated ellipse are found by translation in the same way that the center has been translated. Thus, the vertices are (-3 + 2, 1), (-3 – 2, 1), (-3, 1 + 4), and (- 3, 1 – 4), or, (-1, 1), (-5, 1), (-3, 5), and (-3, 3). The foci are (-3, 1) + 2√3) and (-3, 1 - 2√3).

19. EXAMPLE 3 CONTINUED The graph for the equation (-3, 1) (-3, -3) (-3, 5) (--3, 1-2√3 ) (--3, 1+2√3 )

20. TRY THIS… For each ellipse find the center, vertices and foci, and draw a graph. 1.25x + 9y + 150 x – 36y + 260 = 0 2.9x + 25y – 26x + 150y + 260 = 0

21. CH. 10.3 HOMEWORK Textbook pg. 442 #2, 6, 8, 12, 16, 20 & 22