The tangent PT and the normal PN to the parabola y 2 = 4 a x at a point P on

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Equation of tangent and normal at point $P (a t^{2}, 2 a t)$ is $t y = x = a t^{2}$ and $y = - t x + 2 a t + a t^{2}$ Let centroid of $△ PTN$ is $R (h, k)$ . $∴ h = \frac{a t ^{2} + ( - a t ^{2} ) + 2 a + a t ^{2}}{3}$ and $k = \frac{2 a t}{3}$
$\Rightarrow \Rightarrow \Rightarrow 9 k^{2} 3 h 3 h = 4 a (3 h - 2 a) = 2 a + a \cdot (\frac{3 k}{2 a})^{2} = 2 a + \frac{9 k ^{2}}{4 a}$ $∴$ Locus of centroid is $y^{2} = \frac{4 a}{3} (x - \frac{2 a}{3})$ $∴ Vertex (\frac{2 a}{3}, 0)$ ; directrix $x - \frac{2 a}{3} = - \frac{a}{3} \Rightarrow x = \frac{a}{3}$ Latusrectum $= \frac{4 a}{3}$ $∴$ Focus $(\frac{a}{3} + \frac{2 a}{3}, 0)$ i.e. $(a, 0)$

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