$A \equiv\left(\frac{5 r-1}{r+1}, \frac{5 r-1}{r+1}, \frac{10 r+2}{r+1}\right)$
$(\overrightarrow{\mathrm{OQ}} \cdot \overrightarrow{\mathrm{OA}})-\frac{|\overrightarrow{\mathrm{OP}} \times \overrightarrow{\mathrm{OA}}|^{2}}{5}=10$
$\overrightarrow{\mathrm{OQ}} \cdot \overrightarrow{\mathrm{OA}}=\frac{10}{\mathrm{r}+1}(15 \mathrm{r}+1)$
$|\overrightarrow{\mathrm{OP}} \times \overrightarrow{\mathrm{OA}}|^{2}=\frac{\mathrm{r}^{2}}{(\mathrm{r}+1)^{2}}(800)$
so by equation (1)
$\frac{10}{r+1}(15 r+1)-\frac{1}{5} \frac{r^{2}(800)}{(r+1)^{2}}=10$
$2 r^{2}-14 r=0$
$r=7, r \neq 0$