Radical Axis of Two Circles

Locus of points having equal tangent lengths to both circles. Always a straight line perpendicular to the line joining the centres. For intersecting circles, it is the common chord.

Let $S_1 \equiv x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0$ and $S_2 \equiv x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0$.

For a point $P(x, y)$, the tangent lengths to $S_1$ and $S_2$ are $\sqrt{S_1}$ and $\sqrt{S_2}$ respectively (where $S_1, S_2$ denote the expressions evaluated at $P$).

The radical axis is the locus where these are equal:

$$\sqrt{S_1} = \sqrt{S_2} \implies S_1 = S_2$$ $$(x^2 + y^2 + 2g_1x + 2f_1y + c_1) = (x^2 + y^2 + 2g_2x + 2f_2y + c_2)$$

The $x^2$ and $y^2$ terms cancel, giving:

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

This is the equation $S_1 - S_2 = 0$ — a linear equation, confirming the radical axis is a straight line.

Perpendicularity to line of centres: The line of centres has direction $(g_1 - g_2, f_1 - f_2)$ (from $(-g_1,-f_1)$ to $(-g_2,-f_2)$). The radical axis has normal direction $(g_1-g_2, f_1-f_2)$ — they are perpendicular.

Special cases:

• Intersecting circles: radical axis is the common chord
• Tangent circles: radical axis is the common tangent at the point of tangency
• Non-intersecting circles: radical axis lies between the two circles (closer to the smaller one)