2. Ellipses
5.3
(Chapter 10 – Conics)

3. The Ellipse Major Axis Vertices Minor Axis Co-Vertices Center Foci
Orientation

4. Equations

5. Write an Equation Given the Foci and Vertices
Find the value of a.
Find the value of b.
Find the coordinates of the center.
Write equation.

6. Example 1
What is an equation for the ellipse? A. B. C. D.

7. Write an Equation Given the Lengths of the Axes
Write an equation for the ellipse with vertices at (–6, -2) and (4, –2) and co-vertices at (–1, –4) and (–1, 0).

8. Example 2
Write an equation for the ellipse with vertices at (–9, 1) and (1, 1) and co-vertices at (−4, 3) and (− 4, − 1). A. B. C. D.

9. Graph an Ellipse Find the coordinates of the center and foci and the
lengths of the major and minor axes of the ellipse with
equation x2 + 4y2 – 6x – 16y – 11 = 0. Graph the ellipse.
Complete the square for each variable to write in standard form.
x2 + 4y2 – 6x – 16y – 11 = 0 Original equation
x2 – 6x + ■ + 4(y2 – 4y + ■) = 11 + ■ + 4(■) Complete the squares.
(x2 – 6x + 9) + 4(y2 – 4y + 4) = (4)
(x – 3)2 + 4(y – 2)2= Write as perfect squares.
Divide each side by 36.

10. Center: Length of Major Axis: Length of Minor Axis: Foci: c2= a2 – b2
(3, 2)
a2 = 36 so a = 6
→12
b2 = 9 so b = 3 →6
The foci are located at .
c2 = 36 – 9 or 27
c2 = 27

11. Example 3
What are the coordinates of the center and foci and the lengths of the major and minor axes of the ellipse with equation 4x2 + 25y2 + 16x – 150y = 0?
A. center: (–2, 3) major axis: 10 minor axis: 4 foci: (3, – ), (3, –2 – )
B. center: (2, –3) major axis: 10 minor axis: 4 foci: (– , 3), (–2 – , 3)
C. center: (–2, 3) major axis: 10 minor axis: 4 foci: (– , 3), (–2 – , 3)
D. center: (2, –3) major axis: 10 minor axis: 4 foci: (3, – ), (3, –2 – )