Problem 841

Show that

$$\frac{p^{2}}{a^{2}}+\frac{q^{2}}{b^{2}}+\frac{r^{2}}{c^{2}}=\frac{p^{2}+q^{2}+r^{2}}{x^{2}+y^{2}+z^{2}}$$

if

$$\frac{p}{x}=\frac{q}{y}=\frac{r}{z} \quad \text { and } \quad \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{3}}{c^{2}}=1 \text {. }$$