Hyperbola 2 Question 5

5.

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

(a) $45 \sqrt{5}$

(b) $54 \sqrt{3}$

(c) $60 \sqrt{3}$

(d) $36 \sqrt{5}$

(2018 Main)

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Answer:

Correct Answer: 5. (a)

Solution:

  1. Tangents are drawn to the hyperbola $4 x^{2}-y^{2}=36$ at the point $P$ and $Q$.

Tangent intersects at point $T(0,3)$

Clearly, $P Q$ is chord of contact.

$\therefore$ Equation of $P Q$ is $-3 y=36$

$\Rightarrow \quad y=-12$

Solving the curve $4 x^{2}-y^{2}=36$ and $y=-12$,

we get $\quad x= \pm 3 \sqrt{5}$

Area of $\triangle P Q T=\frac{1}{2} \times P Q \times S T=\frac{1}{2}(6 \sqrt{5} \times 15)=45 \sqrt{5}$



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