Parabola 2 Question 16
16. The value of $r$ is
(a) $-\frac{1}{t}$
(b) $\frac{t^{2}+1}{t}$
(c) $\frac{1}{t}$
(d) $\frac{t^{2}-1}{t}$
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Solution:
- PLAN (i) If $P\left(a t^{2}, 2 a t\right)$ is one end point of focal chord of parabola $y^{2}=4 a x$, then other end point is $\frac{a}{t^{2}},-\frac{2 a}{t}$
(ii) Slope of line joining two points $\left(x _1, y _1\right)$ and $\left(x _2, y _2\right)$ is given by $\frac{y _2-y _1}{x _2-x _1}$
If $P Q$ is focal chord, then coordinates of $Q$ will be $\frac{a}{t^{2}},-\frac{2 a}{t}$.
Now, slope of $Q R=$ slope of $P K$
$$ \begin{aligned} & \frac{2 a r+\frac{2 a}{t}}{a r^{2}-\frac{a}{t^{2}}}=\frac{2 a t}{a t^{2}-2 a} \Rightarrow \quad \frac{r+1 / t}{r^{2}-1 / t^{2}}=\frac{t}{t^{2}-2} \\ \Rightarrow \quad & \frac{1}{r-\frac{1}{t}}=\frac{t}{t^{2}-2} \Rightarrow \quad r-\frac{1}{t}=\frac{t^{2}-2}{t}=t-\frac{2}{t} \\ \Rightarrow \quad & r=t-\frac{1}{t}=\frac{t^{2}-1}{t} \end{aligned} $$
