First slide

Ellipse

Question

The minimum distance of any point on the ellipse $x^{2} + 3 y^{2} + 4 x y = 4$ from its centre is

Moderate

A

$2 \sqrt{\sqrt{5} - 2}$
B

$2 \sqrt{\sqrt{5} + 2}$
C

$2 (\sqrt{2} - 1)$
D

$2 (\sqrt{2} + 1)$

Solution

$Let any point on the ellipse be (r \cos θ, r \sin θ)$
$\Rightarrow d^{2} = \frac{4}{\cos^{2} θ + 3 \sin^{2} θ + 2 \sin 2 θ} - \frac{8}{1 + \cos 2 θ + 3 - 3 \cos 2 θ + 4 \sin 2 θ}$
= $\frac{4}{2 \sin 2 θ - \cos 2 θ + 2}$
$H e n c e E_{m i n} = 2 \sqrt{\sqrt{5} - 2}$

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