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Question

Find the equation of an ellipse, the distance between the foci is 8 units and the distance between the directions is 18 units.

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Solution

Let the equation of the ellipse be
x2a2+y2b2=1 ...(i)
We have,
2ae=8 [given]
e=82a
e=4a ...(ii)
Now,
2ae=18 [given]
a=18e2
a=9e ...(iii)
Using equation (ii) and equation (iii), we get
a=9×4a
a2=36
Now,
b2=a2(1e2)
b2=36(ae)2
b2=3616
b2=20
Putting a2=36 in b2=20 in equation (i), we get
x236+y220=1
This is the equation of the required ellipse.


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