1. Graph an equation of an ellipse
EXAMPLE 1
Graph an equation of an ellipse
Graph the equation 4x y2 = Identify the vertices, co-vertices, and foci of the ellipse.
SOLUTION
STEP 1
Rewrite the equation in standard form.
4x y2 = 100
Write original equation.
4x2
100 +
25x2 =
Divide each side by 100. x2 25 +
y24
= 1
Simplify.

2. EXAMPLE 1
Graph an equation of an ellipse
STEP 2
Identify the vertices, co-vertices, and foci. Note that a2 = 25 and b2 = 4, so a = 5 and b = 2. The denominator of the x2 - term is greater than that of the y2 - term, so the major axis is horizontal.
The vertices of the ellipse are at (+a, 0) = (+5, 0). The co-vertices are at (0, +b) = (0, +2). Find the foci.
c2 = a2 – b2 = 52 – = 21,
so c = 21
The foci are at ( , 0), or about ( + 4.6, 0).

3. EXAMPLE 1
Graph an equation of an ellipse
STEP 3
Draw the ellipse that passes through each vertex and co-vertex.

4. GUIDED PRACTICE
for Example 1
Graph the equation. Identify the vertices, co-vertices, and foci of the ellipse. x2 16 +
y29 1.
= 1
SOLUTION
STEP 1
The equation is in standard form. x2 16 +
y29
= 1

5. GUIDED PRACTICE
for Example 1
STEP 2
Equations.
Major Axis
Vertices
Co - vertices x2 16 +
y29
= 1
Horizontal
+ 4, 0
0, + 3
The vertices of the ellipse are at (+ 4, 0) and co-vertices are at (0, + 3). Find the foci.
c2 = a2 – b2 = 42 – = 7,
so c = 7
The foci are at ( , 0).

6. GUIDED PRACTICE
for Example 1
STEP 3
Draw the ellipse that passes through each vertex and co-vertex.

7. GUIDED PRACTICE
for Example 1 x2 36 +
y2 49 2.
= 1
SOLUTION
STEP 1
The equation is in standard form. x2 36 +
y2 49
= 1

8. GUIDED PRACTICE
for Example 1
STEP 2
Equations.
Major Axis
Vertices
Co - vertices x2 36 +
y2 49
= 1
Vertical
0, + 7
+ 6, 0
The vertices of the ellipse are at (0, + 7) and co-vertices are at (+ 6, 0). Find the foci.
c2 = a2 – b2 = y2 – = 13,
so c = 13
The foci are at (0 + , ).

9. GUIDED PRACTICE
for Example 1
STEP 3
Draw the ellipse that passes through each vertex and co-vertex.

10. Rewrite the equation in standard form. 25x2 + 9y2 = 225
GUIDED PRACTICE
for Example 1 3.
25x2 + 9y2 = 225
SOLUTION
STEP 1
Rewrite the equation in standard form.
25x2 + 9y2 = 225
Write original equation.
25x2
225 +
9y2 = 1
Divide each side by 225. x2 9 +
y225
= 1
Simplify.

11. GUIDED PRACTICE
for Example 1
STEP 2
Equations.
Major Axis
Vertices
Co - vertices x2 32 +
y252
= 1
Vertical
(0 + 5),
+ 3, 0
The vertices of the ellipse are at (0 + 5), and co-vertices are at (+ 3, 0). Find the foci.
c2 = a2 – b2 = 25 – 9 = 16
so c =
+ 4
The foci are at (0, + 4).

12. GUIDED PRACTICE
for Example 1
STEP 3
Draw the ellipse that passes through each vertex and co-vertex.