Question:
Find the equation of the circle concentric with the circle $x^{2}+y^{2}+4 x+6 y+11$ = 0 and passing through the point P(5, 4).
Solution:
2 or more circles are said to be concentric if they have the same centre and different radii.
Given, $x^{2}+y^{2}+4 x+6 y+11=0$
The concentric circle will have the equation
$x^{2}+y^{2}+4 x+6 y+c^{\prime}=0$
As it passes through P(5, 4), putting this in the equation
$5^{2}+4^{2}+4 \times 5+6 \times 4+c^{\prime}=0$
$\Rightarrow 25+16+20+24+c^{\prime}=0$
$\Rightarrow c^{\prime}=-85$
The required equation is
$x^{2}+y^{2}+4 x+6 y-85=0$