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Q.
Let an incident ray $L_1=0$ gets reflected at point $A(-2,3)$ on hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ & passes through focus $S (2,0)$, then -
Conic Sections
Solution:
$S \equiv(2,0), S ^{\prime} \equiv(-2,0)$
Using reflection property of hyperbola, S'A $^{\prime}$ is incident ray.
Equation of incident ray
$S^{\prime} A$ is $x=-2$
Equation of reflected ray
SP is $3 x+4 y=6$.
Now $2 ae =4 \Rightarrow ae =2$....(i)
Point $(-2,3)$ lies on hyperbola,
$\therefore \frac{4}{a^2}-\frac{9}{b^2}=1 $
$ \Rightarrow \frac{4}{a^2}-\frac{9}{4-a^2}=1$
on solving it we get $a =4$ (reject), $a =1$....(ii)
$\therefore $ Using (i) & (ii), we get $e=2$
length of latus rectum $=2 a \left(e^2-1\right)=6$
