1. Ellipses When circles go wild. (9.3)

2. POD Give the center and radius for this circle. x 2 + y 2 – 6x – 4y – 12 = 0

3. POD Give the center and radius for this circle. x 2 + y 2 – 6x – 4y – 12 = 0 (x – 3) 2 + (y – 2) 2 = 25 Center: (3, 2) Radius = 5 Now, rewrite the equation so that the right side equals 1.

4. POD Now, rewrite the equation so that the right side equals 1. In general: What patterns do you notice?

5. On to ellipses Hand’s on to start with. Using paper and styrofoam, we’ll make something interesting. Be careful with my materials! In the end, what have we made?

6. On to ellipses Based on what you’ve just done, how would you describe what an ellipse is?

7. Ellipses– the parts On your handout, identify the following parts of the ellipse: center vertices (one is called a “vertex”) foci (one is called a “focus) major and minor axises (2a and 2b) major and minor radii (a and b) x- radius and y-radius focal radii (c)

8. Ellipses– the relationships How many radii do we have in the ellipse? There is a special relationship between them. Let’s take a look on the board.

9. Ellipses– the relationships How many radii do we have in the ellipse? There is a special relationship between them. Let’s take a look on the board. c 2 + b 2 = a 2, or c 2 = a 2 - b 2.

10. Ellipses– the equation Let’s look at the final equation we derived for a circle in the POD. What letters represent the center and the radius?

11. Ellipses– the equation The equation for an ellipse is much the same What letters represent the center and the radius? What does it look like if it’s centered on the origin?

12. Use it Just like with circles, we complete the square to get answers. We also have to divide so the right side equals 1. Find the focal radius and sketch the graph of this ellipse.

13. Use it Start by completing the square. Remember how to do it if the coefficients are not 1?

14. Use it Then divide. Can you tell what the center is? What about the three radii? Now, sketch it.

15. Use it Now, sketch it. Since the x-radius is greater, this is a horizontal ellipse and the x-radius is a. center (2, -5) x-radius (a)= 3 y-radius (b)= 2 Since c 2 + b 2 = a 2, focal radius = √5