QJEEAdv14Maeafe: JEE Advanced 2014 Mathematics - Class 11 JEEAdv14Maeafe Solution: Let be nonzero real numbers. Let and be distinct points on the…

Given, four points $\mathrm{P}\left(a t^2, 2 a t\right), \mathrm{Q}, \mathrm{R}\left(a r^2, 2 a r\right)$ and $S\left(a s^2, 2 a s\right)$ be distinct points on the parabola $y^2=4 a x$ .

Also given PQ is the focal chord of parabola

$\begin{aligned}& y^2=4 a x \\& \therefore \quad \mathrm{Q}=\left(\frac{a}{t^2},-\frac{2 a}{t}\right)\end{aligned}$

Given, $\mathrm{K}=(2 a, 0)$ and PK is parallel to QR

$\therefore$ Slope of line PK $=$ Slope of line QR

$\begin{aligned}& \frac{2 a t-0}{a t^2-2 a}=\frac{2 a r+\frac{2 a}{t}}{a r^2-\frac{a}{t^2}} \\\Rightarrow & \frac{t}{t^2-2}=\frac{r+\frac{1}{t}}{r^2-\frac{1}{t^2}} \\\Rightarrow & \frac{t}{t^2-2}=\frac{r+\frac{1}{t}}{\left(r+\frac{1}{t}\right)\left(r-\frac{1}{t}\right)} \\\Rightarrow & \frac{t}{t^2-2}=\frac{1}{r-\frac{1}{t}} \\\Rightarrow & t r-1=t^2-2 \\\Rightarrow & \quad r=\frac{t^2-1}{t}\end{aligned}$

Hint:

(i) If PQ is a focal chord of parabola $y^2=4 a x$ and $\mathrm{P}=\left(a t^2, 2 a t\right)$ , then $\mathrm{Q}=\left(\frac{a}{t^2},-\frac{2 a}{t}\right)$ .

(ii) If two lines are parallel, then their slopes are equal.

(iii) The slope of line passing through two points $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ is $\frac{y_2-y_1}{x_2-x_1}$ .