Find center and radius of the circle
$2 x^{2} + 2 y^{2} - 4 x - 6 y + 1 = 0$

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Solution

Given,

$2 x^{2} + 2 y^{2} - 4 x - 6 y + 1 = 0$

$\Rightarrow x^{2} + y^{2} - 2 x - 3 y + \frac{1}{2} = 0$

$2 g x = - 2 x \to g = - 1$

$2 f y = - 3 y \to f = - \frac{3}{2}$

$r = \sqrt{g^{2} + f^{2} - c} = \sqrt{(- 1)^{2} + {(- \frac{3}{2})}^{2} + {(\frac{1}{2})}^{2}} = \frac{\sqrt{7}}{2}$

$∴ 2 x^{2} + 2 y^{2} - 4 x - 6 y + 1 = 0$ is a circle with center $(1, \frac{3}{2})$ and radius $\frac{\sqrt{7}}{2}$

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