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Question
If the latus rectum of an ellipse with axis along x-axis and centre at origin is 10, distance between foci = length of minor axis, then the equation of the ellipse is ______.
Solution
If the latus rectum of an ellipse with axis along x-axis and centre at origin is 10, distance between foci = length of minor axis, then the equation of the ellipse is `x^2/100 + y^2/50` = 1.
Explanation:
Given that `(2b^2)/a` = 10 and 2ae = 2b
⇒ b = ae
Again, we know that b2 = a2(1 – e2)
or 2a2e2 = a2
⇒ e = `1/sqrt(2)` ....(Using b = ae)
Thus `a = bsqrt(2)`
Again `(2b^2)/a` = 10
or b = `5sqrt(2)`.
Thus we get a = 10
Therefore, the required equation of the ellipse is `x^2/100 + y^2/50` = 1
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