The radical axis

Intersecting two circles with a system looks unmanageable (two second-degree equations), but there is a very neat trick that works only between circles.

Property — Radical axis

Let $\gamma_1: x^2 + y^2 + a_1 x + b_1 y + c_1 = 0$ and $\gamma_2: x^2 + y^2 + a_2 x + b_2 y + c_2 = 0$ be two distinct circles. Subtracting their equations term by term gives $(a_1 - a_2)\,x + (b_1 - b_2)\,y + (c_1 - c_2) = 0,$ which is the equation of a line, called the radical axis of the two circles. This line has a fundamental property: it contains all the intersection points of $\gamma_1$ and $\gamma_2$.

Proof

If $P(x_0;y_0)$ belongs to both circles, then both equations are satisfied. Their difference (which is still true) gives exactly the equation of the radical axis evaluated at $P$. Hence every intersection of $\gamma_1$ and $\gamma_2$ lies on the radical axis. ∎

The subtraction eliminates the $x^2$ and $y^2$ terms (identical in the two equations), leaving a first-degree equation: that is why the result is a line.

Links

Topics: Circonferenza analitica
Concepts: Asse radicale · Equazione circonferenza · Intersezione circonferenze
Skills: Dimostrare · Geometria analitica