Dynamic Activity

Ellipses

Take Note Key Concept: Properties of Ellipses with Center (0, 0)

The length of the major axis is 2 a 2 eh  and the length of the minor axis is 2b. For any point P on an ellipse, P F 1 + P F 2 = 2 a . p , f sub 1 , plus p , f sub 2 , equals 2 eh .

Problem 1 Writing an Equation of an Ellipse

What is an equation in standard form of an ellipse centered at the origin with vertex ( − 6 , 0 ) open negative 6 comma 0 close  and co-vertex (0, 3)?

Since one vertex is ( − 6 , 0 ) , open negative 6 comma 0 close comma  the other vertex is (6, 0). The major axis is horizontal. Since one co-vertex is (0, 3), the other co-vertex is ( 0 , − 3 ) . open 0 comma negative 3 close .  The minor axis is vertical. So a = 6 , b = 3 , a 2 = 36 , eh equals 6 comma b equals 3 comma . eh squared , equals 36 comma  and b 2 = 9 . b squared , equals 9 .

x 2 a 2 + y 2 b 2 = 1 Standard form of a horizontal ellipse x 2 36 + y 2 9 = 1 Substitute for  a 2  and  b 2 . table with 2 rows and 3 columns , row1 column 1 , fraction x squared , over eh squared end fraction . plus . fraction y squared , over b squared end fraction , column 2 equals 1 , column 3 cap standardformofahorizontalellipse , row2 column 1 , fraction x squared , over 36 end fraction , plus , fraction y squared , over 9 end fraction , column 2 equals 1 , column 3 cap substitutefor . eh squared , and , b squared , . , end table

✓ Got It?

1. What is the equation in standard form of an ellipse centered at the origin with vertex (0, 5) and co-vertex (2, 0)?