Question

Let $P(at^2, 2at), Q, R(ar^2, 2ar)$ be three points on a parabola $y^2 = 4ax$. If $PQ$ is the focal chord and $PK, QR$ are parallel where the co-ordinates of $K$ is $(2a, 0)$, then the value of $r$ is

• A. $\frac{t}{1-t^2}$
• B. $\frac{1-t^2}{t}$
• C. $\frac{t^2 + 1}{t}$
• D. $\frac{t^2 - 1}{t}$

Answer 1

Answer: (D)

Here, coordinate of $Q$ will be $\left(\frac{a}{t^{2}}, \frac{-2 a}{t}\right)$ :
Slope of $Q R=\frac{2}{r-\frac{1}{t}}$
Slope of $P K=\frac{2 a t}{a t^{2}-2 a}=\frac{2 t}{t^{2}-2}$
Since, Slope of $Q R=$ Slope of $P K$
$\therefore \frac{2}{r-\frac{1}{t}}=\frac{2 t}{t^{2}-2}$
$\Rightarrow r=\frac{t^{2}-1}{t}$