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. (b)
Solution:
- 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 $\triangle P Q T=\frac{1}{2} \times P Q \times S T=\frac{1}{2}(6 \sqrt{5} \times 15)=45 \sqrt{5}$
