Find the coordinate of the centre and radius of the following circle.
$2 x^{2} + 2 y^{2} - 3 x + 5 y = 7$ .

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Solution

we know general equation of circle is

$x^{2} + y^{2} + 2 g x + 2 f y + c = 0$

with centre $(- g, - f)$ and radius $= \sqrt{g^{2} + f^{2} - c}$
consider

$2 x^{2} + 2 y^{2} - 3 x + 5 y = 7$

$x^{2} + y^{2} - \frac{3}{2} x + \frac{5}{2} y - \frac{7}{2} = 0$

centre $(\frac{3}{4}, \frac{- 5}{4})$ radius $= \sqrt{{(\frac{3}{4})}^{2} + {(\frac{- 5}{4})}^{2} + \frac{7}{2}}$

$= \sqrt{\frac{9}{10} + \frac{25}{16} + \frac{7}{2}} = \sqrt{\frac{90}{16}} = \frac{3 \sqrt{10}}{4}$

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