Explanation

(c)
Equation of any tangent to the parabola is
$y = mx + \frac{a}{m}$
[image:11_image18.emf]
It passes through $A(h,\ k)$
$\therefore k = mh + \frac{a}{m}$
$\Rightarrow m^{2}h - mk + a = 0$
Let $m_{1}$ and $m_{2}$ be the roots
$\Rightarrow m_{1} + m_{2} = \frac{k}{h},\ m_{1}m_{2} = \frac{a}{h}$
$\therefore\tan{60{^\circ}} = \left| \frac{m_{1} - m_{2}}{1 + m_{1}m_{2}} \right|$
$\Rightarrow 3 = \frac{\left( m_{1} - m_{2} \right)^{2}}{\left( 1 + m_{1}m_{2} \right)^{2}} \Rightarrow 3 = \frac{\frac{k^{2}}{h^{2}} - \frac{4a}{h}}{\left( 1 + \frac{a}{h} \right)^{2}}$
$\Rightarrow 3(h + a)^{2} = k^{2} - 4ah$
$\therefore$ Locus of a point is
$y^{2} = 3{(x + a)}^{2} + 4ax$