The eccentricity of an ellipse having centre at the origin, axes along the co-ordinate axes and passing through the points (4, 5) and (6, 2) is

x²/a² + y²/b² = 1
Let 1/a² = A, 1/b² = B
Ax² + By² = 1
(4, 5) 16A + 25B = 1 (1)
(6, 2) 36A + 4B = 1 (2)
Subtracting (2) from (1),
-20A + 21B = 0
16A + 25B = 1
Solving these two equations we get:
15A = 3/4 => A = 1/20 => a² = 20
and B = 1/5
=> b² = 5
b² = a²(1 - e²)
5 = 20(1 - e²)
1/4 = 1 - e²
e² = 3/4
e = √3/2
Hence, option C is correct.