Answer
(a) $\frac{b}{a}$
(b) $a=b$
Work Step by Step
(a) Given the coordinates of $A(a,0)$ and $B(0,b)$, we can find the midpoint as $P(\frac{a}{2},\frac{b}{2})$. The slope of line $OP$ with $O(0,0)$ is then given by $m=\frac{b/2-0}{a/2-0}=\frac{b}{a}$
(b) The slope of line $AB$ is given by $m'=\frac{b-0}{0-a}=-\frac{b}{a}$. If $OP$ is perpendicular to $AB$, the product of their slopes should equal to $-1$, thus $mm'=-\frac{b}{a}\times\frac{b}{a}=-1$ or $a^2=b^2$. As $a\gt0, b\gt0$, we have $a=b$ for the two lines to be normal to each other.
