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  4. Let a tangent to the curve y2=24 x meet the curve x y=2 at the points A and B. Then the mid points of such line segments A B lie on a parabola with the

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Q. Let a tangent to the curve $y^2=24 x$ meet the curve $x y=2$ at the points $A$ and $B$. Then the mid points of such line segments $A B$ lie on a parabola with the

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Solution:

$ y ^2=24 x$
$ a =6$
$ xy =2 $
$ AB \equiv ty = x +6 t ^2$.......(1)
$ AB \equiv T = S _1 $
$ kx + hy =2 hk $.........(2)
From (1) and (2)
$\frac{ k }{1}=\frac{ h }{- t }=\frac{2 hk }{-6 t ^2}$
$\Rightarrow$ then locus is $y ^2=-3 x$
Therefore directrix is $4 x=3$