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The normal at a point on the parabola $y^{2} = 4 x$ passes through $(5, 0)$ . If two more normals to this parabola also pass through (5, 0) , then the centroid of the triangle formed by the feet of these three normals is

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$(\frac{1}{2}, \frac{1}{2})$

$(5, 0)$

$(0, 2)$

$(2, 0)$

answer is A.

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Detailed Solution

$G i v e n p a r a b o l a y^{2} = 4 x \Rightarrow a = 1$

Equation of Normal at t is $y = - x t + 2 t + t^{3}$

sub (5, 0) $\Rightarrow 0 = - 5 t + 2 t + t^{3}$

$\Rightarrow t^{3} = 3 t \Rightarrow t = 0, \sqrt{3}, - \sqrt{3}$

$c e n t r i o d = (\frac{{at}_{1}^{2} + {at}_{2}^{2} + {at}_{3}^{2}}{3}, 2 a \frac{(t_{1} + t_{2} + t_{3})}{3}) = (2, 0)$

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