Statement 1: The only circle having radius √10 and a diameter along the line 2x+y=5 is x^2…

Question

Statement 1: The only circle having radius $\sqrt{10}$ and a diameter along the line $2x+y=5$ is $x^2+y^2-6x+2y=0$.
Statement 2: $2x+y=5$ is a normal to the circle $x^2+y^2-6x+2y=0$.

Accepted Answer

Circle: centre $(3,-1)$, radius $\sqrt{9+1}=\sqrt{10}$. Since $2(3)+(-1)=5$, the centre lies on the given line, so the line is indeed a normal --- Statement 2 is true.

However, infinitely many circles of radius $\sqrt{10}$ have centres on the line $2x+y=5$ (any point on that line works as a centre), so the given circle is not the *only* one --- Statement 1 is false.

Solutions · 1