SATHEE: Parabola 2 Question 16

Parabola 2 Question 16

16. The value of $r$ is

(a) $- \frac{1}{t}$

(b) $\frac{t^{2} + 1}{t}$

(d) $\frac{t^{2} - 1}{t}$

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Solution:

PLAN (i) If $P (a t^{2}, 2 a t)$ 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 $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ 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 $P K$

$\begin{aligned} \frac{2 a r + \frac{2 a}{t}}{a r^{2} - \frac{a}{t^{2}}} = \frac{2 a t}{a t^{2} - 2 a} \Rightarrow \frac{r + 1 / t}{r^{2} - 1 / t^{2}} = \frac{t}{t^{2} - 2} \\ \Rightarrow & \frac{1}{r - \frac{1}{t}} = \frac{t}{t^{2} - 2} \Rightarrow r - \frac{1}{t} = \frac{t^{2} - 2}{t} = t - \frac{2}{t} \\ \Rightarrow & r = t - \frac{1}{t} = \frac{t^{2} - 1}{t} \end{aligned}$

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Parabola 2 Question 16

16. The value of r is

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16. The value of $r$ is