Question

9. 4x9y2 36

Answer 1

The given equation of ellipse is

4 x 2 +9 y 2 =36 (1)

This can be further written as,

4 x 2 +9 y 2 =16 x 2 9 + y 2 4 =1

Since the denominator of x 2 9 is greater than the denominator of y 2 4 , major axis is x axis.

Hence the equation of ellipse is represented as

x 2 a 2 + y 2 b 2 =1 (2)

where, a is the length of semi major axis, b is the length of semi minor axis,and c is the distance of the focus from the center of the ellipse which is given as,

c= a 2 − b 2 . (3)

By comparing this equation of ellipse with the given equation of ellipse, we get, a=3 and b=2

By substituting value of a and b in (3) , we get

c= a 2 − b 2 c= 9−4 c= 5

Since major axis is x axis, coordinates of foci are (±c,0)=(± 5 ,0)

Since major axis is x axis, coordinates of vertices are (±a,0)=(±3,0)

Length of major axis = 2a =2×3 =6

Length of minor axis = 2b =2×2 =4

Eccentricity e= c a = 5 3

Length of latus rectum = 2 b 2 a = 2× 2 2 3 = 2×4 3 = 8 3

Thus, the equation 4 x 2 +9 y 2 =36 has foci (± 5 ,0) , vertices (±3,0) ,length of major axis 6, length of minor axis 4, eccentricity 5 3 and length of latus rectum 8 3