Find the coordinates of the centres and the radii of the circles whose equation is:
$3 x^{2} + 3 y^{2} - 5 x - 6 y + 4 = 0$ .

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Solution

Given equation is $3 x^{2} + 3 y^{2} - 5 x - 6 y + 4 = 0$
Divide throughout by 3
$\Rightarrow x^{2} + y^{2} - \frac{5}{3} x - 2 y + \frac{4}{3} = 0$
Now comparing above equation with general equation of circle, $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$
Centre of the circle is $(- g, - f) = (\frac{5}{6}, 1)$
and radius $= \sqrt{g^{2} + f^{2} - c} = \sqrt{\frac{25}{36} + 1 - \frac{4}{3}} = \frac{\sqrt{13}}{6}$

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