Question

If the tangent and normal at the extremities of a focal chord of a parabola intersect at $(x_1,y_1)$ and $(x_2,y_2)$ respectively, then

• A. $x_1=x_2$
• B. $x_1=y_2$
• C. $y_1=y_2$
• D. $x_2=y_1$

Answer 1

Answer: (C)

$y_1=y_2$

Let $P({at}_1^2,2{at}_1),Q({at}_2^2,2{at}_2)$ be a focal chord of the parabola $y^2=4ax.$

The tangents at $P$ and $Q$ meet at $({at}_1{t}_2,a(t_1+t_2)).$

$\therefore x_1={at}_1{t}_2=-a$

$(\because PQ$ is a focal chord $\therefore t_1{t}_2=-1)$

$y_1=a(t_1+t_2)$

Normals at $P,Q$ meets at $(2a+a(t_1^2+t_2^2+t_1{t}_2),-{at}_1{t}_2(t_2+t_2))$

$\therefore x_2=2a+a(t_1^2+t_2^2-1),y_2=-a(-1)(t_1+t_2)=a\left(t_1+t_2\right)$

Clearly $y_1=y_2$