Question

The normal at the point $\left(b t_{1}^{2}, 2 b t_{1}\right)$ on a parabola meets the parabola again in the point $\left(b t_{2}^{2}, 2 b t_{2}\right),$ 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. $t_{2}=t_{1}+\frac{2}{t_{1}}$

Answer 1

Answer: (A)

Key Idea : Equation of normal at a point $\left(a t^{2}, 2 a t\right)$ on $a$ perabola $y^{2}=4 a x$ is $y=-t x+2 a t+b t^{3}$
Equation of the normal at point $\left(b t_{1}^{2}, 2 b t_{1}\right)$ on parabola is
$y=-t_{1} x+2 b t_{1}+b t_{1}^{3}$
It also passes through $\left(b t_{2}^{2}, 2 b t_{2}\right),$ then
$2 b t_{2} =-t_{1} \cdot b t_{2}^{2}+2 b t_{1}+b t_{1}^{3} $
$2 t_{2}-2 t_{1} =-t_{1}\left(t_{2}^{2}-t_{1}^{2}\right) $
$=-t_{1}\left(t_{2}+t_{1}\right)\left(t_{2}-t_{1}\right) $
$\Rightarrow 2 =-t_{1}\left(t_{2}+t_{1}\right)$
$\Rightarrow t_{2}+t_{1} =-\frac{2}{t_{1}} $
$\Rightarrow t_{2} =-t_{1}-\frac{2}{t_{1}}$
Note :
1. If a normal drawn at a point $t_{1}$ on the parabola meets the parabola again at a point $t_{2}$, then
$t_{2}=-t_{1}-\frac{2}{t_{1}}$
2. Three normals can be drawn at a point on a parabola.
3. Foot of the normals drawn from a point are called conormal points.