7. 36x2 4y2 - 144

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Solution

The given equation of ellipse is

36 x 2 +4 y 2 =144 (1)

This can be further written as,

36 x 2 +4 y 2 =144 x 2 4 + y 2 36 =1

Since the denominator of y 2 36 is greater than the denominator of x 2 4 , major axis is y axis.

Therefore, the equation of ellipse is represented as

x 2 b 2 + y 2 a 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=6 and b=2

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

c= a 2 − b 2 c= 36−4 c= 32 c=4 2

Since major axis is y axis, coordinates of foci are (0,±c)=(0,±4 2 )

Since major axis is y axis, coordinates of vertices are (0,±a)=(0,±6)

Length of major axis = 2a =2×6 =12

Length of minor axis = 2b =2×2 =4

Eccentricity e= c a = 4 2 6 = 2 2 3

Length of latus rectum = 2 b 2 a = 2× 2 2 6 = 2×4 6 = 8 6 = 4 3

Thus, the equation 36 x 2 +4 y 2 =144 has foci (0,±4 2 ) , vertices (0,±6) ,length of major axis 12, length of minor axis 4, eccentricity 2 2 3 and length of latus rectum 4 3

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$I f e_{1} i s t h e e c c e n t r i c i t y o f t h e c o n i c 9 x^{2} + 4 y^{2} = 36 a n d e^{2} i s t h e e c c e n t r i c i t y o f t h e c o n i c 9 x^{2} - 4 y^{2} = 36, t h e n$

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