Let the tangent at $$A(p$$,$$q)$$ to the ellipse $$x216+y24=1$$ form a triangle of least area with the coordinate axes (p$$,q>$$0) . Then $$p2+q2$$ is equal to

T:$$x4cos$$⁡θ$$+y2sin$$⁡θ$$=1$$ Area $$=12(4sec$$⁡θ$$)(2cosec$$⁡θ$$)$$ Area $$=8sin$$⁡$$2$$θ For area to be least θ$$=$$π$$4$$ . ∴$$p=22$$ and $$q=2$$∴$$p2+q2=8+2=10$$

Ellipse

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Question

$Let the tangent at A (p, q) to the ellipse \frac{x^{2}}{16} + \frac{y^{2}}{4} = 1 form a triangle of least area with the coordinate axes (p, q > 0) .$
$Then (p^{2} + q^{2}) is equal to$

Moderate

Solution

$\begin{array}{l} T : \frac{x}{4} \cos θ + \frac{y}{2} \sin θ = 1 \\ Area = \frac{1}{2} (4 \sec θ) (2 cosec θ) \\ Area = \frac{8}{\sin 2 θ} \\ For area to be least θ = \frac{π}{4} . \\ ∴ p = 2 \sqrt{2} and q = \sqrt{2} \\ ∴ (p^{2} + q^{2}) = 8 + 2 = 10 \end{array}$

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