Tangents are drawn to the hyperbola $4 x^{2} - y^{2} = 36$ at the points $P a n d Q$ If these tangents Intersect at the point $T (0, 3)$ then the area (in sq.units) of $Δ P T Q$ is

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$60 \sqrt{3}$

$45 \sqrt{5}$

$36 \sqrt{5}$

$54 \sqrt{3}$

answer is B.

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Detailed Solution

Let PQ is chord of contact w.r.t point $T (0, 3),$ then
Eqation of PQ is $4 (0 \times x) - (y \times 3) = 36$ or $y = - 12 (S_{1} = 0)$
Solving line PQ with the hyperbola, we get
$4 x^{2} - 144 = 36 \Rightarrow 4 x^{2} = 180 \Rightarrow x^{2} = 45 \Rightarrow x = \pm 3 \sqrt{5}$
$∴ P (- 3 \sqrt{5}, - 12), Q (3 \sqrt{5}, - 12)$
$∴$ Area of triangle $T P Q = \frac{1}{2} \times P Q \times T M = \frac{1}{2} \times \sqrt{5} \times 15 = 45 \sqrt{5} s q . u n i t s$