Calculus, Thomas, 13th ed, Chapter 1.1, Problem 12

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

$$P=(p,\sqrt{p})=(1/a^2,1/a).$$

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