The Locus

Let $p = OA$ and $q = OB$ . Then segment $AB$ is on the line through points $(0, q)$ and $(p, 0)$ , which is $y = -\frac{q}{p}x + q$ , and $PQ$ is on the line through $(p, q)$ and a slope of $\frac{p}{q}$ , which is $y = \frac{p}{q}x + q - \frac{p^2}{q}$ . These two lines intersect at $Q(\frac{p^3}{p^2 + q^2}, \frac{q^3}{p^2 + q^2})$ , so the locus of $Q$ is defined by $x = \frac{p^3}{p^2 + q^2}$ and $y = \frac{q^3}{p^2 + q^2}$ . Since $p^2 + q^2 = b^2$ by Pythagorean's Theorem on $\triangle ABO$ , $x = \frac{p^3}{b^2}$ and $y = \frac{q^3}{b^2}$ .

This means that $x^{\frac{2}{3}} + y^{\frac{2}{3}}$ $=$ $(\frac{p^3}{b^2})^{\frac{2}{3}} + (\frac{q^3}{b^2})^{\frac{2}{3}}$ $=$ $\frac{p^2 + q^2}{b^{\frac{4}{3}}}$ $=$ $\frac{b^2}{b^{\frac{4}{3}}}$ $=$ $b^{\frac{2}{3}}$ . Therefore, the locus of $Q$ can be defined as $\boxed{x^{\frac{2}{3}} + y^{\frac{2}{3}} = b^{\frac{2}{3}}}$