Find the equation of a system of circles which have the line $x - y = 0$ for their radical axis.
Equation of system of circles is
$x^2 + y^2 + 2gx + 2fy + c + \lambda \left(px + qy + r\right) = 0$ $\;\;\; \cdots \; (1)$
where $\;$ $\lambda$ $\;$ is a constant and
$px + qy + r = 0$ $\;\;\; \cdots \; (2)$ $\;$ is the radical axis.
Given: $\;$ Radical axis: $\;$ $x - y = 0$ $\;\;\; \cdots \; (3)$
Since equations $(2)$ and $(3)$ represent the same line,
coefficients of the $x$ and $y$ terms and the constant term must be identical.
i.e. $\;$ $p = 1, \; q = -1, \; r = 0$
$\therefore \;$ Substituting the values of $p$, $q$ and $r$ in equation $(1)$, the required equation of system of circles is
$x^2 + y^2 + 2gx + 2fy + c + \lambda \left(x - y\right) = 0$
i.e. $\;$ $x^2 + y^2 + \left(2g + \lambda\right) x + \left(2f - \lambda\right) y + c = 0$