If the pair of lines $$a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$$ intersect on the $$y$$ axis then

The pair of lines represented by
$$3a{x^2} + 5xy + \left( {{a^2} - 2} \right){y^2} = 0$$ are perpendicular to each other for

Prove that, in an ellipse, the perpendicular from a focus upon any tangent and the line joining the centre of the ellipse to the point of contact meet on the corresponding directrix.

Prove that $$\cos \,ta{n^{ - 1}}\sin \,{\cot ^{ - 1}}x = \sqrt {{{{x^2} + 1} \over {{x^2} + 2}}} $$.