Thank you for reporting, we will resolve it shortly
Q.
Point $R(h, k)$ divides a line segment between the axes in the ratio $1: 2$. Then, the equation of the line is
Straight Lines
Solution:
Let equation of line $A B$ is
$\frac{x}{a}+\frac{y}{b}=1.....$(i)
Let a point $R(h, k)$ divide line $A B$ in the ratio $1: 2$.
By using section formula, (internal division)
$R(h, k)=\left(\frac{1 x_2+2 x_1}{1+2}, \frac{1 y_2+2 y_1}{1+2}\right)$
$ \Rightarrow h=\frac{1 \times 0+2 \times a}{1+2} $,
$k=\frac{1 \times b+2 \times 0}{1+2} $
$[\because P(x, y) $ divide the line $\left(x_1, y_1\right) $ and $\left(x_2, y_2\right)$ in the ratio $ m: n $ internally $]$
$ \left.\therefore P(x, y)=\left(\frac{m x_2+n x_1}{n+m}, \frac{m y_2+n y_1}{n+m}\right)\right] $
$ \Rightarrow h=\frac{2 a}{3}, k=\frac{b}{3}$
$ \Rightarrow a=\frac{3 h}{2}, b=3 k$
On putting the values of $a$ and $b$ in Eq. (i), we get
$\frac{x}{\frac{3 h}{2}}+\frac{y}{3 k}=1 $
$ \frac{2 x}{3 h}+\frac{y}{3 k}=1$
$ \frac{2 k x+h y}{3 h k}=1 $
$ 2 k x+h y=3 h k $
