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Area between Two Curves

P Q are t...

Question

$P$ & $Q$ are the points of contact of the tangents drawn from the point $T$ to the parabola $y^{2} = 4 a x .$ If $P Q$ be the normal to the parabola at $P$ , then show that $T P$ is bisected by the directrix.

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Solution

Let $P (a t_{1}^{2}, 2 a t_{1}) & Q (a t_{2}^{2}, 2 a t_{2})$ on $y^{2} = 4 a x$
co-ordinate of $T (a t_{1} t_{2}, a (t_{1} + t_{2}))$ which is point of intersection of tangent at $P$ & $Q$ equation of $P Q$ which is normal at $P$
$y + t_{1} x = 2 a t_{1} + a t_{1}^{3} . . . (1)$
equatioin of PQ is
$(t_{1} + t_{2}) y = 2 x + 2 a t_{1} t_{2} . . . (2)$
equation (1) & (2) are same
Compare slope $\frac{2}{t_{1} + t_{2}} = - t_{1}$
$\Rightarrow t_{1}^{2} + t_{1} t_{2} = - 2$
Now mid point of $T P$
$x = \frac{a t_{1}^{2} + a t_{1} t_{2}}{2} = \frac{a (t_{1}^{2} + t_{1} t_{2})}{2}$
$x = \frac{a (- 2)}{2} = - a$
$x = - a$ which is directrix
Hence $T P$ bisect the directrix.

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