Distance of Radical Axis from the First Centre

Distance of Radical Axis from the First Centre

For circles:

$$x^2+y^2+2g_1x+2f_1y+c_1=0$$

and:

$$x^2+y^2+2g_2x+2f_2y+c_2=0$$

their radical axis is:

$$2(g_1-g_2)x+2(f_1-f_2)y+(c_1-c_2)=0$$

The distance from centre $$O_1(-g_1,-f_1)$$ is:

$$d=\frac{|2(g_1-g_2)(-g_1)+2(f_1-f_2)(-f_1)+c_1-c_2|}{2\sqrt{(g_1-g_2)^2+(f_1-f_2)^2}}$$

Usage

- Used when the distance of the common chord from a circle's centre is required directly from the equations.