First say that $\left( {a,0} \right)\,\& \,\left( {0,b} \right)$ are the endpoints of the segment on the x-axis and on the y-axis respectively.
Because the line segment has constant length $\sqrt {a^2 + b^2 } = c\;,\;c>0$.
If $m=n$ then $P$ is the midpoint of the segment, so $P:\left( {\frac{a}{2},\frac{b}{2}} \right)$.
As $\left| b \right| \to c\quad \Rightarrow \quad \left| a \right| \to 0$ and visa versa.