Find the equation of the ellipse whose foci are `(2,3),(-2,3)` and whose semi-minor axes is `sqrt5`.

Explanation

Here `S -= (2, 3), S' -= (-2, 3), b = sqrt(5)`
`rArr SS' = 4 rArr 2ae = 4`
`rArr ae = -2`
But `b^(2) = a^(2) (1-e^(2)) = a^(2) - 4 rArr a = 3 `
Let `P(alpha, beta)` by any point on the ellipse, then
`SP + S'P = 2a = 6`
`rArrsqrt((alpha-2)^(2)+(beta-3)^(2))+sqrt((alpha+2)^(2)+(beta-3)^(2))=6`
After simplification it can be reduced to `5alpha^(2) + 9beta^(2) - 54y + 36 = 0`
Hence locus of `(alpha, beta)` is
`5x^(2) + 9y^(2) - 54y + 36 = 0`
Which is the required equation of the ellipse