A line of fixed lengthlmoves so that its ends are on the coordinate axes. Determine the locus ofPon this line which divides it in the ratio m:n. What is the locus if m=n?

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- Aug 24th 2008, 12:02 PMkatie_lamAnalytic Geometry Locus
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? - Aug 24th 2008, 01:55 PMPlato
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.