Question

Let a circle $C:(x-h{)}^2+(y-k{)}^2=r^2,k>0$, touch the $x$-axis at $(1,0)$. If the line $x+y=0$ intersects the circle $C$ at $P$ and $Q$ such that the length of the chord $PQ$ is $2$ , then the value of $h+k+r$ is equal to ______.

Answer 1

Answer: 7

$k=r$
$h=1$
$OP=r,PR=1$
$OR=\left|\frac{r+1}{\sqrt{2}}\right|$

$r^2=1+\frac{(r+1{)}^2}{2}$
$2r^2=2+r^2+1+2r$
$r^2-2r-3=0$
$(r-3)(r+1)=0$
$r=3,-1$
$h+k+r=1+3+3$
$=7$