Find the minimum distance of any point on the curve $x^{2} + y^{2} + 2 x y = 8$ from the origin.

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Solution

$x^{2} + y^{2} + 2 x y = 8$
$\Rightarrow (x + y)^{2} = 8 \Rightarrow x + y = 2 \sqrt{2} \Rightarrow x + y - 2 \sqrt{2} = 0$ .... $(1)$
Now, the minimum distance from $(0, 0)$ to this line
$∣ ∣ ∣ ∣ ∣ \frac{0 + 0 - 2 \sqrt{2}}{\sqrt{{(1)}^{2} + {(1)}^{2}}} ∣ ∣ ∣ ∣ ∣ = \frac{2 \sqrt{2}}{\sqrt{2}} = 2$

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