Radical axis and intersections

Problem

Find the radical axis of $\gamma_1: x^2+y^2-2x=0$ and $\gamma_2: x^2+y^2-2y-3=0$ and their intersection points, if any.

Solution

Radical axis: we subtract term by term $\gamma_1 - \gamma_2$: $(-2x) - (-2y - 3) = 0 \iff -2x + 2y + 3 = 0 \iff \boxed{2x - 2y - 3 = 0.}$ We obtain $x = y + \tfrac{3}{2}$.

System with $\gamma_1$: we substitute into $x^2 + y^2 - 2x = 0$: $\left(y + \tfrac{3}{2}\right)^2 + y^2 - 2\left(y + \tfrac{3}{2}\right) = 0 \iff 2y^2 + y - \tfrac{3}{4} = 0 \iff 8y^2 + 4y - 3 = 0.$ $\Delta = 16 + 96 = 112$, hence $y = \dfrac{-4 \pm \sqrt{112}}{16} = \dfrac{-1 \pm \sqrt{7}}{4}.$

With $x = y + \tfrac{3}{2} = \dfrac{5 \pm \sqrt{7}}{4}$, the intersection points are $\boxed{\left(\tfrac{5+\sqrt{7}}{4};\ \tfrac{-1+\sqrt{7}}{4}\right) \ \text{e}\ \left(\tfrac{5-\sqrt{7}}{4};\ \tfrac{-1-\sqrt{7}}{4}\right).}$ The two circles therefore intersect at two distinct points (consistently with $\Delta > 0$).

Links

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