Statement 1: The normals at three points P ,Q ,R of the parabola $y^{2} = 4 a x$ meet in (h,k), then centroid of $Δ P Q R$ lies on x=0.
Statement 2: If A,B,C are three co-normal points of the parabola $y^{2} = 4 a x$ , then centroid of $Δ A B C$ lies on axis of the parabola.

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Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation of Statement 1

Statement 1 is true, Statement 2 is false

Statement 1 is true, Statement 2 is true; Statement 2 is the correct explanation of Statement 1

Statement 1 is false, Statement 2 is true

answer is D.

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Detailed Solution

Let $P (a t_{1}^{2}, 2 a t_{1}), Q (a t_{2}^{2}, 2 a t_{2}), R (a t_{3}^{2}, 2 a t_{3})$ be the three points on parabola.
Equation of the normals at ‘t’ is $y + x t = 2 a t + a t^{3}$
The above normal is passing through (h,k)
$\Rightarrow k + h t = 2 a t + a t^{3} \Rightarrow a t^{3} + (2 a - h) t - k = 0$
The above is a cubic equation in ‘t’ whose roots are $t_{1}, t_{2}, t_{3} \Rightarrow \sum t_{1} = 0$
Centroid of $Δ P Q R i s (\frac{a (t_{1}^{2} + t_{2}^{2} + t_{3}^{2})}{3}, \frac{2 a}{3} (t_{1} + t_{2} + t_{3})) = (\frac{a (t_{1}^{2} + t_{2}^{2} + t_{3}^{2})}{3}, 0)$
Centroid lies on y=0
Hence statement-1 is false but statement-2 is true.