Calculus TA

Problem A point $P$ in the first quadrant lies on the graph of the function $f(x)=\sqrt{x}$. Express the coordinates of $P$ as functions of the slope of the line joining $P$ to the origin. Answer Let $p$ be the $x$-coordinate of $P$ and $a$ be the slope of the line joining $P$ to the origin. Then  $$a=\frac{\sqrt{p}-0}{p-0}=\frac{1}{\sqrt{p}},$$ $$\sqrt{p}=1/a,\;\;p=1/a^2.$$ Hence, the coordinates of $P$ is $$P=(p,\sqrt{p})=(1/a^2,1/a).$$