The center is at (1,-1), Transverse axis is parallel to x-axis; Length of latus rectum is 9, distance between directrices = 8/√(13)
d = a^2/c => 2d = 2a^2/c = 8/√(13) (d is distance of one directrix from centre)
p = b^2/a => 2p = 2b^2/a = 9 ( p is semi-latus rectum)
A basic relation between semi-major axes, a and b, and c, eccentricity is b^2 = c^2 – a^2
Using b^2 = c^2 – a^2 and the previous two equations,
b^2 = c^2 – a^2 => 9a = 2b^2 = 2c^2 – 2a^2
using a^2 = 4c/√(13), => c^2 = 13a^4/16
Therefore, 9a = 13a^4/8 – 2a^2
13a^4 – 16a^2 – 72a = 0
13a^3 – 16a – 72 = 0 (since a ≠ 0)
This factorises as,
(a – 2)(13a^2 + 26a + 36) = 0
Only 1 real root: a = 2
From c^2 = 13a^4/16 => c^2 = 13, c= √(13)
From 2a^2 = 8c/√(13) = 8 a^2 = 4, a = 2 (confirmation)
From b^2 = c^2 – a^2, => b^2 = 13 – 4 = 9, b = 3
Eqn of ellipse
x^2/a^2 – y^2/b^2 = 1
x^2/4 – y^2/9 = 1 (Centred at (0,0))
or,
(x – 1)^2/4 – (y + 1)^2/9 = 1 (Centred at (1,-1))