Solve p(a)/p(10-a)quadp(a)=a(10-a)/a-5

$$\frac { p ( a ) } { p ( 10 - a ) } \quad p ( a ) = \frac { a ( 10 - a ) } { a - 5 }$$

Solve for a

$$\left\{\begin{matrix}a=0\text{, }&p\neq 0\\a=\frac{5\left(\sqrt{p\left(p+8\right)}+p-4\right)}{2\left(p-1\right)}\text{; }a=\frac{5\left(-\sqrt{p\left(p+8\right)}+p-4\right)}{2\left(p-1\right)}\text{, }&p\leq -8\text{ or }\left(p\neq 1\text{ and }p>0\right)\\a=\frac{20}{3}\approx 6.666666667\text{, }&p=1\end{matrix}\right.$$

Solve for p

$$\left\{\begin{matrix}p=\frac{\left(a-10\right)^{2}}{a\left(a-5\right)}\text{, }&a\neq 5\text{ and }a\neq 10\text{ and }a\neq 0\\p\neq 0\text{, }&a=0\end{matrix}\right.$$