A line of fixed length l moves so that its ends are on the coordinate axes. Determine the locus of P on this line which divides it in the ratio m:n. What is the locus if m=n?
Here are some hints.
First say that $\displaystyle \left( {a,0} \right)\,\& \,\left( {0,b} \right)$ are the endpoints of the segment on the x-axis and on the y-axis respectively.
Because the line segment has constant length $\displaystyle \sqrt {a^2 + b^2 } = c\;,\;c>0$.
If $\displaystyle m=n$ then $\displaystyle P$ is the midpoint of the segment, so $\displaystyle P:\left( {\frac{a}{2},\frac{b}{2}} \right)$.
As $\displaystyle \left| b \right| \to c\quad \Rightarrow \quad \left| a \right| \to 0$ and visa versa.