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  4. Tangents to the parabola y2=4ax at P(a t12 , 2 a t1) and Q(a t22 , 2 a t2) meet at T. If triangle PTQ is right-angled at T, then (1/P S)+(1/Q S) is equal to (where, S is the focus of the given parabola)

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Q. Tangents to the parabola $y^{2}=4ax$ at $P\left(a t_{1}^{2} , 2 a t_{1}\right)$ and $Q\left(a t_{2}^{2} , 2 a t_{2}\right)$ meet at $T.$ If $\triangle PTQ$ is right-angled at $T,$ then $\frac{1}{P S}+\frac{1}{Q S}$ is equal to (where, $S$ is the focus of the given parabola)

NTA AbhyasNTA Abhyas 2020Conic Sections

Solution:

$T$ lies on the directrix
and $PQ$ is the focal chord.
$\Rightarrow t_{1}t_{2}=-1$
$\frac{1}{P S}=\frac{1}{a \left(1 + t_{1}^{2}\right)}$ , $\frac{1}{Q S}=\frac{1}{a \left(1 + t_{2}^{2}\right)}=\frac{t_{1}^{2}}{a \left(1 + t_{1}^{2}\right)}$
So, $\frac{1}{P S}+\frac{1}{Q S}=\frac{1}{a}$