JEE Advanced 2014 · Paper 2 · Mathematics Q51

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Let $a$ , $r$ , $s$ , $t$ be nonzero real numbers. Let $P(at^2,2at)$ , $Q$ , $R(ar^2,2ar)$ and $S(as^2,2as)$ be distinct points on the parabola $y^2=4ax$ . Suppose that $PQ$ is the focal chord and lines $QR$ and $PK$ are parallel, where $K$ is the point $(2a,0)$ .

The value of $r$ is

-\dfrac1t

\dfrac{t^2+1}t

\dfrac1t

\dfrac{t^2-1}t

Answer and worked solution

Official answer: D

Focal chord. If $P$ has parameter $t$ , the other end Q has parameter $-\frac1t$ .

Slopes. For chord through parameters $t_1,t_2$ , the slope is $\dfrac2{t_1+t_2}$ . Also
$m_{PK}=\frac{2at}{at^2-2a}=\frac{2t}{t^2-2}.$

Parallel.
$\frac2{r-\frac1t}=\frac{2t}{t^2-2}\implies r-\frac1t=\frac{t^2-2}t\implies r=\frac{t^2-1}t.$

(D)

Topics:Parabola

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JEE Advanced 2014 · Paper 2, Mathematics question 51