Example — intersection of two circles

Example

Intersect $\gamma: x^2 + y^2 - 2x - 4y + 4 = 0$ and $\gamma': x^2 + y^2 - 4x = 0$.

Radical axis: $(-2 - (-4))x + (-4 - 0)y + (4 - 0) = 0 \iff 2x - 4y + 4 = 0 \iff x - 2y + 2 = 0$.

Line-circle system (we choose $\gamma'$, which is simpler): $\begin{cases} x = 2y - 2 \\ x^2 + y^2 - 4x = 0 \end{cases}$ Substituting: $(2y-2)^2 + y^2 - 4(2y-2) = 0 \iff 5y^2 - 16y + 12 = 0$.

$\Delta = 256 - 240 = 16$, hence $y = \dfrac{16\pm 4}{10}$: $y_1 = 2$, $y_2 = \tfrac{6}{5}$.

Points: $\boxed{G_1(2;2) \ \text{e}\ G_2\!\left(\tfrac{2}{5};\tfrac{6}{5}\right).}$

Once the values of $y$ are obtained from the second-degree equation, the abscissae are recovered from the equation of the radical axis $x = 2y - 2$.

Links

Topics: Circonferenza analitica
Concepts: Asse radicale · Intersezione circonferenze · Intersezione retta circonferenza
Methods: Circonferenza intersezione
Skills: Geometria analitica · Risolvere sistemi