# Locus of a point on a variable line.

A variable line passing through a point $(6,\,6)$ cuts the coordinate axes at the point $A$ and $B$. If the point $P$ divides $AB$ internally in the ratio $2:1$ what is the locus of the point $P$.

My attempt: Let $P$ have coordinates $(h,\,k)$ By section formula $h=\frac{2a}{3}$ $k=\frac{b}{3}$ But, I cannot find a way to eliminate $a$ and $b$. How should I use the information that the line passes through $(6,\,6)$.

As you have assumed $x$- intercept $a$ and $y$-intercept $b$, equation of line in intercept form is :

$$\frac xa + \frac yb =1$$

Since it passes through the point $(6,6)$

$$\frac 6a + \frac 6b =1$$

Now put the value of $a$ and $b$ in terms of $h$ and $k$;

$$\frac {6}{3h/2} + \frac {6}{3k} =1$$

Now simplify and replace $h \rightarrow x$ and $k\rightarrow y$

So, desired locus is -

$$4y+2x-xy=0$$