Find the locus of mid points of the chord to the circle $(x-3)^2 + (y-2)^2 =1$ passing through the point $(3,7)$.
I have a doubt , the point $(3,7)$ lies outside the circle isn't? What does passing through the point $(3,7)$ mean? Sorry if I am wrong. Just give me clues.
Best Answer
Let $M(x,y)$ one of points on the locus, $A(3,7)$ and $B(3,2)$ be a center of the circle.
Thus, since $BM\perp MA$, we see that our locus is placed on the circle: $$\left(x-\frac{3+3}{3}\right)^2+\left(y-\frac{7+2}{2}\right)^2=\left(\frac{AB}{2}\right)^2$$ or $$(x-3)^2+(y-4.5)^2=6.25$$