Question

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

• A. $2kx+hy=3hk$
• B. $2kx+hy=2hk$
• C. $2kx-by-3hk$
• D. None of these

Answer 1

Answer: (A)

Let equation of line $AB$ is $\frac{x}{a}+\frac{y}{b}=1\dots \left(i\right)$
A point $R\left(h,k\right)$ divides the line $AB$ in the ratio $1:2$.
Now, $R\left(h,k\right)=\left(\frac{1x_2+2x_1}{1+2},\frac{1y_2+2y_1}{1+2}\right)$
$\Rightarrow h=\frac{\left(1\times 0\right)+\left(2\times a\right)}{1+2},k=\frac{\left(1\times b\right)+\left(2\times 0\right)}{1+2}$
$\Rightarrow h=\frac{2a}{3}$, $k=\frac{b}{3}$
$\Rightarrow a=\frac{3h}{2}$, $b=3k$
On putting the values of $a$ and $b$ in $\left(i\right)$, we get



$\frac{x}{\frac{3h}{2}}+\frac{y}{3k}=1$
$\Rightarrow \frac{2x}{3h}+\frac{y}{3k}=1$
$\Rightarrow 2kx+hy=3hk$