Question

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. $t_2 = - \left( t_1 + \frac{2}{t_1} \right)$
• D. None of these

Answer 1

Answer: (C)

Let the normal at $‘t_1’$ cuts the parabola again at the point '$t_2$'. the equation of the normal
at $(at_1{^2}, 2at_1)$ is
$y + t_1x = 2at_1 + at_1{^3}$
Since it passes through the point '$t_2$'
i.e $(at_2{^2}, 2at_2)$
$ \therefore \, \, 2at_{2} + at_{1}t_{2} ^{2} = 2 at_{1} + at_{1} ^{3}$
$ \Rightarrow 2a\left(t_{1} - t_{2} \right) + at_{1}\left(t_{1} ^{2} - t_{2} ^{2}\right) = 0$
$ \Rightarrow 2 + t_{1} \left(t_{1} + t_{2}\right) = 0$
$ \left(\because t_{1} - t_{2 } \ne0\right)$
$ \Rightarrow 2 + t_{1} ^{2} + t_{1 } t_{2} = 0 $
$ \Rightarrow t_{1} t_{2} = -\left(t_{1} ^{2} + 2 \right) $
$\Rightarrow t_{2} = - \left( t_{1} + \frac{2}{t_{1}}\right) $