EXAMPLE 1 Graph an equation of an ellipse

EXAMPLE 1 Graph an equation of an ellipse Graph the equation 4 x 2 + 25 y 2 = 100. Identify the vertices, co-vertices, and foci of the ellipse. SOLUTION STEP 1 Rewrite the equation in standard form. 4 x 2 + 25 y 2 = 100 Write original equation. 4 x 2 25 x 2 = 100 + 100 y 2 x 2 25 + 4 = 1 Divide each side by 100. Simplify.

EXAMPLE 1 Graph an equation of an ellipse STEP 2 Identify the vertices, co-vertices, and foci. Note that a 2 = 25 and b 2 = 4, so a = 5 and b = 2. The denominator of the x 2 - term is greater than that of the y 2 - 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. c 2 = a 2 – b 2 = 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. 2 y 2 x 1. 16 + 9 = 1 SOLUTION STEP 1 The equation is in standard form. y 2 x 2 16 + 9 = 1

GUIDED PRACTICE for Example 1 STEP 2 Equations. Major Axis Vertices Co - vertices y 2 x 2 Horizontal + 4, 0 0, + 3 + = 1 16 9 The vertices of the ellipse are at (+ 4, 0) and co-vertices are at (0, + 3). Find the foci. c 2 = a 2 – b 2 = 42 – 32 = 7, so c = 7 The foci are at ( + 7 , 0).

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

GUIDED PRACTICE 2. for Example 1 y 2 x 2 36 + 49 = 1 SOLUTION STEP 1 The equation is in standard form. y 2 x 2 36 + 49 = 1

GUIDED PRACTICE STEP 2 Equations. y 2 x 2 36 + 49 = 1 for Example 1 Major Axis Vertical Vertices Co - vertices 0, + 7 + 6, 0 The vertices of the ellipse are at (0, + 7) and co-vertices are at (+ 6, 0). Find the foci. c 2 = a 2 – b 2 = y 2 – 62 = 13, so c = 13 The foci are at (0 + , 13 ).

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

GUIDED PRACTICE for Example 1 3. 25 x 2 + 9 y 2 = 225 SOLUTION STEP 1 Rewrite the equation in standard form. 25 x 2 + 9 y 2 = 225 Write original equation. 25 x 2 9 y 2 = 1 225 + 225 y 2 x 2 9 + 25 = 1 Divide each side by 225. Simplify.

GUIDED PRACTICE for Example 1 STEP 2 Equations. Major Axis Vertices Co - vertices y 2 x 2 Vertical (0 + 5), + 3, 0 + = 1 2 2 3 5 The vertices of the ellipse are at (0 + 5), and co-vertices are at (+ 3, 0). Find the foci. c 2 = a 2 – b 2 = 25 – 9 = 16 so c = + 4 The foci are at (0, + 4).

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