If one of the diameters of the circle x^2+y^2-2x-6y+6=0 is a chord of another circle 'C', …

Question

If one of the diameters of the circle $x^2+y^2-2x-6y+6=0$ is a chord of another circle 'C', whose centre is at $(2,1)$, then its radius is

Answer: 3

Accepted Answer

Given circle: centre $(1,3)$, $r^2=1+9-6=4\Rightarrow r=2$. Its diameter, as a chord of $C$, is bisected by $C$'s centre's perpendicular foot at $(1,3)$ itself (the midpoint of any diameter is the circle's own centre).

$$d=\text{dist}((2,1),(1,3))=\sqrt{1+4}=\sqrt5,\qquad R^2=d^2+r^2=5+4=9\Rightarrow R=3.$$

Solutions · 1