Centre and radius of several circles (II)

Text

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

• (a) $x^2+y^2-3x+y-2=0$
• (b) $x^2+y^2+\dfrac{3}{2}x+\dfrac{8}{3}y=0$
• (c) $x^2+y^2-2x+y-10=0$

Solution

With $C\left(-\tfrac{a}{2};-\tfrac{b}{2}\right)$ and $R=\sqrt{\tfrac{a^2}{4}+\tfrac{b^2}{4}-c}$:

(a) $a=-3,\ b=1,\ c=-2$: $\ C\left(\tfrac32;-\tfrac12\right)$, $\ R^2=\tfrac94+\tfrac14+2=\tfrac92$, $\ R=\sqrt{4.5}\approx2.12$. (b) $a=\tfrac32,\ b=\tfrac83,\ c=0$: $\ C\left(-\tfrac34;-\tfrac43\right)$, $\ R^2=\tfrac{9}{16}+\tfrac{16}{9}=\tfrac{337}{144}$, $\ R=\dfrac{\sqrt{337}}{12}\approx1.53$. (c) $a=-2,\ b=1,\ c=-10$: $\ C\left(1;-\tfrac12\right)$, $\ R^2=1+\tfrac14+10=\tfrac{45}{4}$, $\ R=\dfrac{3\sqrt5}{2}\approx3.35$.

$\boxed{\ (a)\ C\left(\tfrac32;-\tfrac12\right),\ R\approx2.12;\quad (b)\ C\left(-\tfrac34;-\tfrac43\right),\ R\approx1.53;\quad (c)\ C\left(1;-\tfrac12\right),\ R=\tfrac{3\sqrt5}{2}\ }$