Locus of point of intersection of normal drawn at end points of focal chord of a parabola $y^{2} = 4

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Locus of point of intersection of normal drawn at end points of focal chord of a parabola $y^{2} = 4ax$ is :

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Sol. [Key Concept: For a focal chord $t_{1}t_{2} = - 1$; intersect the normals at its ends and eliminate the parameter to get $y^{2} = a(x - 3a)$.]$$t_{1}t_{2} = - 1$$$$\begin{matrix} \Rightarrow \ & k = a\left( t_{1} + t_{2} ight) \ h & \ = a\left( t_{1}^{2} + t_{2}^{2} + 1 ight) \end{matrix}$$$$\begin{matrix} \frac{k^{2}}{a^{2}} & \ = \left( \frac{h}{a} - 1 ight) + 2( - 1) = \frac{h}{a} - 3 \ \Rightarrow \ y^{2} & \ = a(x - 3a) \end{matrix}$$

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