Two circles, centres, radii and intersections

Text

Consider the two circles $\gamma_1:\ x^2+y^2+x-2y-5=0,\qquad \gamma_2:\ x^2+y^2-3x+4y-8=0.$

• (a) find their centres;
• (b) find their radii;
• (c) draw them;
• (d) find their intersections and check they match the drawing.

Solution

(a)–(b) From centre $\left(-\tfrac a2,-\tfrac b2\right)$ and $r=\sqrt{\tfrac{a^2}{4}+\tfrac{b^2}{4}-c}$: $C_1\!\left(-\tfrac12;1\right),\quad r_1=\sqrt{\tfrac14+1+5}=\sqrt{6.25}=2.5;$ $C_2\!\left(\tfrac32;-2\right),\quad r_2=\sqrt{\tfrac94+4+8}=\sqrt{14.25}\approx 3.775.$

(d) Subtracting the two equations gives the radical axis (line of the common points): $4x-6y+3=0\ \Rightarrow\ x=\frac{6y-3}{4}.$ Substituting into $\gamma_1$: $52y^2-44y-83=0$, with $\Delta=19200$, $\sqrt{19200}=80\sqrt3$: $y=\frac{44\pm80\sqrt3}{104}=\frac{11\pm20\sqrt3}{26}\ \Rightarrow\ y_1\approx 1.755,\quad y_2\approx -0.909.$ The corresponding $x$: $x_1\approx 1.883$, $x_2\approx -2.114$. Intersections $\approx(1.88;\,1.76)$ and $(-2.11;\,-0.91)$.

$\boxed{C_1(-\tfrac12;1),\ r_1=2.5;\ C_2(\tfrac32;-2),\ r_2\approx3.775;\ \text{2 intersections}}$