Centre and radius of several circles (I)

Text

Find the centre and radius of the following circles, then draw them.

• (a) $x^2+y^2-x+3y-9=0$
• (b) $x^2+y^2-\dfrac{3}{2}x-16=0$
• (c) $x^2+y^2+2x+4y-4=0$
• (d) $x^2+y^2-2x+6y=0$

Solution

From the standard form $x^2+y^2+ax+by+c=0$ one reads $C\left(-\tfrac{a}{2};-\tfrac{b}{2}\right)$ and $R=\sqrt{\tfrac{a^2}{4}+\tfrac{b^2}{4}-c}$.

(a) $a=-1,\ b=3,\ c=-9$: $\ C\left(\tfrac{1}{2};-\tfrac{3}{2}\right)$, $\ R^2=\tfrac14+\tfrac94+9=\tfrac{23}{2}$, $\ R=\sqrt{11.5}\approx3.39$. (b) $a=-\tfrac32,\ b=0,\ c=-16$: $\ C\left(\tfrac34;0\right)$, $\ R^2=\tfrac{9}{16}+16=\tfrac{265}{16}$, $\ R=\dfrac{\sqrt{265}}{4}\approx4.07$. (c) $a=2,\ b=4,\ c=-4$: $\ C(-1;-2)$, $\ R^2=1+4+4=9$, $\ R=3$. (d) $a=-2,\ b=6,\ c=0$: $\ C(1;-3)$, $\ R^2=1+9=10$, $\ R=\sqrt{10}\approx3.16$.

$\boxed{\ (a)\ C\left(\tfrac12;-\tfrac32\right),\ R\approx3.39;\quad (b)\ C\left(\tfrac34;0\right),\ R\approx4.07;\quad (c)\ C(-1;-2),\ R=3;\quad (d)\ C(1;-3),\ R=\sqrt{10}\ }$