Question

Point $R(h,k)$ divides a line segment between the axes in the ratio $1:2$. Find equation of the line.

Answer 1

Let the respective coordinates of $A$ and $B$ be $(x,0)$ and $(0,y)$, since point $R(h,k)$ divides $AB$ in the ratio $1:2$.

According to the section formula,

$x=\frac{m{x}_2+n{x}_1}{m+n}$

$\Rightarrow (h,k)=\{\frac{1\times 0+2\times x}{1+2},\frac{1\times y+2\times 0}{1+2}\}$

$\Rightarrow (h,k)=\{\frac{2x}{3},\frac{y}{3}\}$

$\Rightarrow h=\frac{2x}{3}$ and $k=\frac{y}{3}$

$\Rightarrow x=\frac{3h}{2}$ and $y=3k$

Therefore ,the respective coordinates of $A$ and $B$ are $\{\frac{3h}{2},0\}$ and $(0,3k)$

Now the equation of line $AB$ passing through points$\{\frac{3h}{2},0\}$ and $(0,3k)$ is $(y-0)=\frac{3k-0}{0-\frac{3h}{2}}\{x-\frac{3h}{2}\}$

$\Rightarrow y=-\frac{2k}{h}\{x-\frac{3h}{2}\}$

$\Rightarrow hy=-2kx+3hk$

$\Rightarrow 2kx+hy=3hk$