0
$\begingroup$

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 $(6,\,6)$.

$\endgroup$
1
$\begingroup$

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 -

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.