Find the equation of the circle concentric with
$x^{2} + y^{2} - 4 x - 6 y - 3 = 0$
and which touches the y-axis.

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Solution

The given equation of circle is
$x^{2} + y^{2} - 4 x - 6 y - 3 = 0 \dots (i)$
$∴$ centre is $(- g, - f) = (2, 3)$
The required circle is concentric with (i)
so, they have same centre = (2, 3)
Also, the required circle touches y - axis at A
$∴$ CA = radius = 2
$∴$ equation of circle is
$(x - 2)^{2} + (y - 3)^{2} = 4$
$\Rightarrow x^{2} + y^{2} - 4 x - 6 y + 9 = 0$

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