Question

Let $P$ and $Q$ are the points on the parabola $y^2=4x,$ if so that the line segment $PQ$ subtends a right angle at the vertex. If $PQ$ intersects the axis of the parabola at $R$, then the distance of the vertex from $R$ is

• A. $1$
• B. $2$
• C. $4$
• D. $6$

Answer 1

The correct option is C

$4$

Explanation for the correct option.

Finding the distance of the vertex from$R$

It shows in the figure that slope of $PX$ and $QX$ are perpendicular to each other,

so product of slope of both lines will be equal to $-1$

$$\begin{gathered} \frac{2t}{t^2}\times \frac{2m}{m^2}=-1 \\ tm=-4 \end{gathered}$$

$\begin{array}{rcl}PQ:\frac{x-t^2}{t^2-m^2}& =& \frac{y-2t}{2t-2m}\\ \frac{\alpha -t^2}{(t+m)(t-m)}& =& \frac{0-2t}{2(t-m)}\\ & \Rightarrow & \alpha -t^2=-t^2-tm\\ & \Rightarrow & \alpha =4\end{array}$$\left[\mathrm{let}R(\alpha ,0)\right]$

Hence, the correct option is (C)