Circle equation from centre and radius

Text

Write the equation of the following circles and draw them.

• (a) centre $C(3;4)$ and radius $4$;
• (b) centre $C(0;1)$ and radius $3$, and centre $D(1;0)$ and radius $2$;
• (c) centre $C(-3;1)$ and radius $2$.

Solution

Start from the “vertex” form $(x-x_0)^2+(y-y_0)^2=R^2$ and expand.

(a) $(x-3)^2+(y-4)^2=16 \implies x^2+y^2-6x-8y+9=0$. (b) $x^2+(y-1)^2=9 \implies x^2+y^2-2y-8=0$; and $(x-1)^2+y^2=4 \implies x^2+y^2-2x-3=0$. (c) $(x+3)^2+(y-1)^2=4 \implies x^2+y^2+6x-2y+6=0$.

$\boxed{\ (a)\ x^2+y^2-6x-8y+9=0;\quad (b)\ x^2+y^2-2y-8=0,\ \ x^2+y^2-2x-3=0;\quad (c)\ x^2+y^2+6x-2y+6=0\ }$