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Latus Rectum

The normal a ...

Question

The normal a point $(b t_{1}^{2}, 2 b t_{1})$ on a parabola meets the parabola again in the point $(b t_{2}^{2}, 2 b t_{2})$ then

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Solution

The correct option is B
Let $P (t_{1}) . Q (t_{2}) & R (t_{3})$
Equation of a normal to $y^{2} = 4 a x i s y + t x = 2 a t + a t^{3}$
This passes through $(h, k) \Rightarrow k + t h = 2 a t + a t^{3} \Rightarrow a t^{3} + (2 a - h) t - k = 0$
$t_{1}, t_{2}, t_{3}$ are the roots of this equation $t_{1} + t_{2} + t_{3} = 0$
Centroid of $Δ$ PQR is $G [\frac{a}{3} (t_{1}^{2} + t_{2}^{2} + t_{3}^{2}), \frac{2 a}{3} (t_{1} + t_{2} + t_{3})]$
$t_{1} + t_{2} + t_{3} = 0 \Rightarrow \frac{2 a}{3} (t_{1} + t_{2} + t_{3}) = 0 \Rightarrow$ G lies on y =0

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