Question

A variable line through the point $P(2,1)$ meets the axes at $A$ and $B.$ Find the locus of the centroid of triangle $OAB$ (where $O$ is the origin).

Answer 1

Let the equation of a line passing through $P(2,1)$ having intercepts a on the $x-$ axis and $b$ on the $y-$ axis is given by

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

And since it contains $(2,1)$

$\therefore$ $\frac{2}{a}+\frac{1}{b}=1⟶\left(1\right)$

the centorid of $\Delta OAB$ is

$(h,k)=(\frac{a}{3},\frac{b}{3})$

i.e. $h=\frac{a}{3}⟶\left(2\right)$ and $k=\frac{b}{3}⟶\left(3\right)$

Using $\left(2\right)$ & $\left(3\right)$ and substituting in $\left(1\right)$ we get

$\frac{2}{3h}+\frac{1}{3k}=1$

$\Rightarrow \frac{2}{h}+\frac{1}{k}=3$

Replacing $(h,k)$ with $(x,y)$ we get the locus of centroid to be $\frac{2}{x}+\frac{1}{y}=3$