The minimum distance between the circle $x^{2} + y^{2} = 9$ and the curve $2x^{2} + 10y^{2} + 6xy =

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The minimum distance between the circle $x^{2} + y^{2} = 9$ and the curve $2x^{2} + 10y^{2} + 6xy = 1$ is:

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[Key Concept: Minimum distance between a circle and a conic = (distance between centres) $-$ (radii); reduce the conic to standard form first.]$$\begin{matrix} r^{2}\left( 2\cos^{2}	heta + 10\sin^{2}	heta + 6sin	heta cos	heta ight) = 1 \ r^{2} = \frac{1}{3sin2	heta - 4cos2	heta + 6} \leq \frac{1}{6 - 5} = 1 \ r \leq 1 \end{matrix}$$Minimum distance between curves $= 3 - 1 = 2$.

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