Question

Let $a,r,s$ and $t$ be non –zero real numbers. Let $P(a{t}^2,2at),Q,R(a{r}^2,2ar)$ and $S(a{s}^2,2as)$ 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 point $(2a,0)$.
The value of $r$ is

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

Answer 1

The correct option is D $\frac{t^2-1}{t}$

(i) $P(a{t}^2,2at)$ is one end point of focal chord of parabola $y^2=4ax$, then other end point is$(\frac{a}{t^2},-\frac{2a}{t})$
(ii) Slope of line joining two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $\frac{y_2-y_1}{x_2-x_1}$
If $PQ$ is focal chord, then coordinates of $Q$ will be
$(\frac{a}{t^2},-\frac{2a}{t})$
Now, slope of $QR$ =slope of $PK$
$\frac{2ar+\frac{2a}{t}}{a{r}^2-\frac{a}{t^2}}=\frac{2at}{a{t}^2-2a}\Rightarrow \frac{r+\frac{1}{t}}{r^2-\frac{1}{t^2}}=\frac{t}{t^2-2}\Rightarrow \frac{1}{r-\frac{1}{t}}=\frac{1}{t^2-2}\Rightarrow r-\frac{1}{t}=\frac{t^2-2}{t}=t-\frac{2}{t}\Rightarrow r=t-\frac{1}{t}=\frac{t^2-1}{t}$