I have a set of circumferences $x^2 + y^2 + \alpha_1 x + \beta_1 y + \gamma_1 + k(x^2 + y^2 + \alpha_2 x + \beta_2 y + \gamma_2) = 0$ $\alpha_1, \alpha_1, \beta_1, \beta_2, \gamma_1, \gamma_2$ given.
I need to find the value of $k$ that correspond to a circumference with center on the line $y = mx$ I tried to set: $x_C = - \frac{\alpha_1 + k\alpha_2}{2}$ and $y_C = - \frac{\beta_1 + k\beta_2}{2}$ and the solve in $k$: $y_C = m x_C$
Am i doing it right?
Thank you, regards.