EXAMPLE 1 Graph an equation of an ellipse

Graph the equation 4x2 + 25y2 = 100. Identify the vertices, co-vertices, and foci of the ellipse.

SOLUTION

STEP 1

Rewrite the equation in standard form.

4x2 + 25y2 = 100 Write original equation.

4x2

100 + 25x2

100100 100= Divide each side by 100.

x2

25 +y2

4 = 1 Simplify.

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 – 22 = 21,

so c = 21

The foci are at ( + 21 , 0), or about ( + 4.6, 0).

EXAMPLE 1 Graph an equation of an ellipse

STEP 3

Draw the ellipse that passes through each vertex and co-vertex.

GUIDED PRACTICE for Example 1

Graph the equation. Identify the vertices, co-vertices, and foci of the ellipse.

1. x2

16 +y2

9 = 1

SOLUTION

STEP 1The equation is in standard form. x2

16 +y2

9 = 1

GUIDED PRACTICE for Example 1

STEP 2

The vertices of the ellipse are at (+ 4, 0) and co-vertices are at (0, + 3). Find the foci.

c2 = a2 – b2 = 42 – 32 = 7,

so c = 7The foci are at ( + 7 , 0).

Equations. Major Axis Vertices Co - vertices

x2

16 +y2

9 = 1 Horizontal + 4, 0 0, + 3

GUIDED PRACTICE for Example 1

STEP 3

Draw the ellipse that passes through each vertex and co-vertex.

GUIDED PRACTICE for Example 1

2. x2

36 + y2

49 = 1

SOLUTION

STEP 1The equation is in standard form.

x2

36 + y2

49 = 1

GUIDED PRACTICE for Example 1

STEP 2

The vertices of the ellipse are at (0, + 7) and co-vertices are at (+ 6, 0). Find the foci.

c2 = a2 – b2 = y2 – 62 = 13,

so c = 13The foci are at (0 + , 13 ).

Equations. Major Axis Vertices Co - vertices

Vertical 0, + 7 + 6, 0 x2

36 + y2

49 = 1

GUIDED PRACTICE for Example 1

STEP 3

Draw the ellipse that passes through each vertex and co-vertex.

GUIDED PRACTICE for Example 1

3. 25x2 + 9y2 = 225

SOLUTION

STEP 1

Rewrite the equation in standard form.

25x2 + 9y2 = 225 Write original equation.

Divide each side by 225.

x2

9 +y2

25 = 1 Simplify.

25x2

225 + 9y2

225 = 1

GUIDED PRACTICE for Example 1

STEP 2

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).

Equations. Major Axis Vertices Co - vertices

x2

32 +y2

52 = 1 Vertical (0 + 5), + 3, 0

GUIDED PRACTICE for Example 1

STEP 3

Draw the ellipse that passes through each vertex and co-vertex.