Question

A variable line through the point $(6/5,6/5)$ cuts the coordinate axes in the points $A$ and $B$. If the point $P$ divides $AB$ internally in the $2:1$, ratio show that the equation to the locus of $P$ is $5xy=2(2x+y)$.

Answer 1

Let the line be $x/a+y/b=1$ and since it passes through the point
$(\frac{6}{5},\frac{6}{5})\therefore \frac{1}{a}+\frac{1}{b}=\frac{5}{6}....\left(1\right)$
It meets the axes at $A(a,0)$ and $B(0,b)$
Let $(h,k)$ be the point which divides $AB$ in the ratio $2:1$ then
$h=\frac{2.0+1.a}{3}=\frac{a}{3},k=\frac{2.b+1.0}{3}=\frac{2b}{3}$
$a=3h,b=3k/2$ Putting in (1) we get
$\frac{1}{3h}+\frac{2}{3k}=\frac{5}{6}$
Generalising $(h,k)$ the locus is
$\frac{y+2x}{3xy}=\frac{5}{6}or2(y+2x)=5xy$