1. Warm-Up
Write the standard equation for an ellipse with foci at (-5,0) and (5,0) and with a major axis of 18. Sketch the graph.

2. 10.4 Day 2 Ellipses
What is the standard equation for an ellipse if the vertex has been translated?

3. Standard Equation of a Translated Ellipse
Horizontal Major Axis:
(x – h)2 a2
(y – k)2 b2 +
= 1
a2 > b2
a2 – b2 = c2
length of major axis: 2a length of minor axis: 2b

4. Standard Equation of a Translated Ellipse
Vertical Major Axis:
(x – h)2 b2
(y – k)2 a2 +
= 1
a2 > b2
a2 – b2 = c2
length of major axis: 2a length of minor axis: 2b

5. (x – 2)2
(y – 1)2 9
Graph +
= 1 16
SOLUTION
STEP 1
Compare the given equation to the standard forms of equations of hyperbolas. The equation’s form tells you that the graph is a hyperbola with a vertical transverse axis. The center is at (h, k) = (2, 1). Because a2 = 16 and b2 = 9, you know that a = 4 and b = 3.
STEP 2
Plot the center, vertices and co-vertices. Vertices are at (h + a ,k) are (6,1) and ( – 2,1)and co-vertices are at (h,k +b) are (2,4) and (2,– 2)
STEP 3
Draw the ellipse that panes through each vertices and co-vertices

6. Write an equation of the ellipse with foci at (1, 2) and (7, 2) and co-vertices at (4, 0) and (4, 4).
SOLUTION
STEP 1
Determine the form of the equation. First sketch the ellipse. The foci lie on the major axis, so the axis is horizontal. The equation has this form:
(x – h)2 a2 +
(y – k)2 b2
= 1
STEP 2
Identify h and k by finding the center, which is halfway between the foci (or the co-vertices)
(h, k) = 2 ) ( ,
= (4, 2)
STEP 3
Find b, the distance between a co-vertex and the center (4, 2), and c, the distance between a focus and the center. Choose the co-vertex (4, 4) and the focus (1, 2): b = | 4 – 2 | = 2 and c = | 1 – 4 | = 3.

7. STEP 4
Find a. For an ellipse, a 2 = b 2 + c 2 = = 13, so a = 13
ANSWER
The standard form of the equation is
(x – 4)2 13 +
(y – 2)2 4
= 1

8. Find the center which is halfway between the vertices.
Write an equation of the ellipse with foci at (3,5) and (3,−1) and vertices at (3,6) and (3,−2).
Plot the given points and make a rough sketch. The ellipse has a vertical axis, so its equation is in the form of: -6 -4 -2 2 4 6
Find the center which is halfway between the vertices.
Find a and c
a = 6−2=4 c = 5−2=3
Find b using b2= a2−c2

9. What is the standard equation for an ellipse if the vertex has been translated?

10. 8.4 Assignment day 2
p. 512, even
p. 531, 7, 10, 17, 18