Find the value $a$ if $2 x^{2} + a y^{2} - 3 x + 2 y - 1 = 0$ represent a circle and also find its radius.

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Solution

Given equation to represents a circle, coefficient of $x^{2}$ and $y^{2}$ must be same i.e. $a = 2$
So the given equation becomes
$2 x^{2} + 2 y^{2} - 3 x + 2 y - 1 = 0$
$\Rightarrow x^{2} + y^{2} - \frac{3}{2} x + y - \frac{1}{2} = 0$
Comparing with general equation of circle, $g = - \frac{3}{4}, f = \frac{1}{2}, c = - \frac{1}{2}$
Hence radius $= \sqrt{g^{2} + f^{2} - c} = \sqrt{\frac{9}{16} + \frac{1}{4} + \frac{1}{2}} = \frac{\sqrt{21}}{4}$

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