Find the equation of ellipse whose foci are (2, 3), (-2, 3) and whose semi major axis is of length \(\sqrt{5}\).

Solution :

Here S = (2, 3) & S’ is (-2, 3) and b = \(\sqrt{5}\) \(\implies\) SS’ = 4 = 2ae \(\implies\) ae = 2

but \(b^2\) = \(a^2(1-e^2)\) \(\implies\) 5 = \(a^2\) – 4 \(\implies\) a = 3

Hence the equation to major axis is y = 3.

Centre of ellipse is midpoint of SS’ i.e. (0, 3)

\(\therefore\) Equation to ellipse is \(x^2\over a^2\) + \({(y-3)}^2\over b^2\) = 1 or \(x^2\over 9\) + \({(y-3)}^2\over 5\) = 1