Question

Two given straight lines meet in $O$, and through a given point $P$ is drawn a straight line to meet them in $Q$ and $R$; if the parallelogram $OQSR$ be completed find the equation to the locus of $R$.

Answer 1

Let the point $S$ be $(x_1,y_1)$ and the fixed point $P$ be $(h,k)$ and the axes be along $OQ$ and $OR$ and coordinates of $OQ$ $OR$ are $(a,0)$ and $(0,b)$ respectively.

As the parallelogram $OQSR$ is completed. So the equation of line $QR$ is

$\frac{x}{a}+\frac{y}{b}=1$

It also passes through the point $P$ whose coordinates are $(h,k)$

$\Rightarrow \frac{h}{a}+\frac{k}{b}=1$

As the coordinates of $S$ are $(x_1,y_1)$

$\frac{h}{x_1}+\frac{k}{y_1}=1$

Generalising the equation,

$\Rightarrow \frac{h}{x}+\frac{k}{y}=1$

is the required locus of $S$