If the length of the latus rectum of an ellipse with major axis along x-axis and centre at origin is 20 units, distance between foci is equal to length of minor axis, then find the equation of the ellipse.

Explanation

It is given that `(2b^(2))/(a) = 20`
`rArr (b^(2))/(a) = 10`
Also, distance between the foci = length of minor axis
i.e., 2c = 2b
`rArr c = b`
`rArr ae = b " " …(i) " " [therefore e = (c )/(a) rArr c = ae]`
Again, we know that
`b^(2) = a^(2) (1-e^(2))`
or `a^(2)e^(2) = a^(2)e^(2) " "` [ From (i)]
`rArr 2a^(2)e^(2) " " rArr e = (1)/(sqrt(2))`
Thus, `a = bsqrt(2)`
Again, `(b^(2))/(a) = 10`
`rArr = (b^(2))/(bsqrt(2))= 10" " rArr b = 10sqrt(2)`
`rArr b^(2) = 200 " " rArr a = 20 " " [therefore a = bsqrt(2)]`
`rArr a^(2) = 400`
Therefore, the required equation of the ellipse is `(x^(2))/(400) + (y^(2))/(200) =1`