Mathematics | VITEEE 2015 | Discussion Page

The normal at the point $(at_{1}^{2},2at^{}_{1})$ on the parabola meets the parabola again in the point

$(at_{2}^{2},2at^{}_{2})$ , then

A) $t_{2}=-t_{1}+\frac{2}{t_{1}}$

B) $t_{2}=-t_{1}-\frac{2}{t_{1}}$

C) $t_{2}=t_{1}-\frac{2}{t_{1}}$

D) $t_{2}=t_{1}+\frac{2}{t_{1}}$

Option B

Equation of the normal at point $(at_{1}^{2},2at^{}_{1})$ on parabola is

$y=-t_{1}x+2at_{1}+at^{3}_{1}$

It also passes through $(at_{2}^{2},2at^{}_{2})$

So, $2at_{2}=-t_{1}(at^{2}_{2})+2at_{1}+at^{3}_{1}$

$\Rightarrow 2t_{2}-2t_{1}=-t_{1}(t^{2}_{2}-t^{2}_{1})$

$\Rightarrow t_{1}+t_{2}=\frac{-2}{t_{1}}$

$\Rightarrow t_{2}=-t_{1}-\frac{2}{t_{1}}$

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