Mathematics | VITEEE 2016 | Discussion Page

The normal at the point ( $at_{1}^{2}, 2at_{1}$) on the parabola, cuts the parabola again at the point whose parameter is

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) none of these

Option C

Let the normal at 't₁' cuts the parabola again at the point 't₂' , the equation of the normal at ( $at_{1}^{2}, 2at_{1}$) is y+t₁ x =2at₁ +at₁³

Since it passes through the point 't₂' i.e,

( $at_{2}^{2}, 2at_{1}$)

$\therefore$ $2at_{2}+at_{1}t_2^2=2at_{1}+at_1^3$

$\Rightarrow2a(t_{1}-t_{2})+at_1(t_1^2-t_2^2)=0$

$\Rightarrow 2+t_{1}(t_{1}+t_{2})=0(\because t_{1}-t_{2}\neq0)$

$\Rightarrow 2+t_{1}^{2}+t_{1}t_{2}=0$

$\Rightarrow t_{1}t_{2}=-(t_{1}^{2}+2)\Rightarrow t_{2}=-\left(t_{1}+\frac{2}{t_{1}}\right)$

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