Find the length of the intercepts of the circle $3 x^{2} + 3 y^{2} - 5 x + 3 y = 0$ on the co-ordinates axes.

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Solution

$\begin{matrix} 3 x^{2} + 3 y^{2} - 5 x + 3 y = 0 \Rightarrow x^{2} + y^{2} - \frac{5}{3} x + \frac{3}{3} y = 0 s tan d e r d e q u a t i o n o f c i r c l e \Rightarrow x^{2} + y^{2} + 2 g x + 2 f y + C = 0 C = 0, g = - \frac{5}{6}, f = \frac{1}{2} T h e n i n t e r c e p t s m a d e b y c o - o r d i n a t e s a r e 2 \sqrt{g^{2} - c} a n d 2 \sqrt{f^{2} - c} \Rightarrow 2 \sqrt{(- \frac{5}{6})} = 2 \times \frac{5}{6} = \frac{5}{3} i n t e r c e p t i o n x - a x i s \end{matrix}$
$\begin{matrix} a n d i n t e r c e p t o n y - a x i s 2 \sqrt{{(\frac{1}{2})}^{2} - 0} = 2 \times \frac{1}{2} = 1 \end{matrix}$

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