Radius of a circle

Question

I'm having trouble with a question where I'm given two points, (-5,-2) & (1,0). Find the equation of the circle. I've used the midpoint formula to get the center which is (x+2) & (y+1) If I'm correct. I used the distance formula to get the full equation, the radius anyway but seem to get it wrong? My answer is root 40/2. Because the r is squared, root 40/2 becomes 10^2....Right? Please correct me If I'm wrong, which I think I am. The answer is apparently 9 :p

Answer 1

If you are only given two points and the circle is deemed determined, then those two points define a diameter. The center is the midpoint $(-2,-1)$, and the radius is the half-distance, $\frac12\sqrt{6^2+2^2}=\sqrt{10}$.

Hence $$\color{blue}{(x+2)^2+(y+1)^2=10}.$$

Now, if $R=9$ is also a given, you can form the parametric equation of the bisector of the two points, $$x=-2-2t\\y=-1+6t$$ and find the two possible centers at distance $9$ from $(1,0)$ along it: $$(-2-2t-1)^2+(-1+6t-0)^2=9^2,$$ or $$40t^2=71,$$ $$t=\pm\sqrt{\frac{71}{40}}.$$ The possible circle equations are $$\color{blue}{(x+2\pm2\sqrt{\frac{71}{40}})^2+(y+1\mp6\sqrt{\frac{71}{40}})^2=81}.$$