Explanation

(a)
Equation of any tangent to the given ellipse is
$y = mx \pm \sqrt{a^{2}m^{2} + b^{2}}$
$\Rightarrow \ y - mx = \pm \sqrt{a^{2}m^{2} + b^{2}}$ …(i)
Equation of perpendicular line is
$my + x = \lambda$
It passes through the centre (0, 0)
$\therefore\ \lambda = 0$
$\therefore my + x = 0$ …(ii)
On squaring and adding Eqs. (i) and (ii)
$y^{2} + m^{2}x^{2} + m^{2}y^{2}x^{2} = a^{2}m^{2} + b^{2}$
$\left( 1 + m^{2} \right)\left( x^{2} + y^{2} \right) = a^{2}m^{2} + b^{2}$
$\Rightarrow \ \left( 1 + \frac{x^{2}}{y^{2}} \right)\ \left( x^{2} + y^{2} \right) = \frac{a^{2}x^{2}}{y^{2}} + b^{2}$
$\Rightarrow \ (x^{2} + {y^{2})}^{2} = a^{2}x^{2} + b^{2}y^{2}$
But $\left( x^{2} + y^{2} \right)^{2} = lx^{2} + my^{2}$
$\therefore\ l = a^{2},\ m = b^{2}$