If the portion of a line intercepted between the coordinates axes is divided by the point $(2,-1)$ in the…
If the portion of a line intercepted between the coordinates axes is divided by the point $(2,-1)$ in the ratio of $3: 2$, then the equation of that line is
$5 x-2 y-20=0$
$2 x-y-5=0$
$3 x-y-7=0$
$x-3 y-5=0$
Solution
Let the equation of the line be
$
\frac{x}{a}+\frac{y}{b}=1
$
The line meets the coordinate axes at $A(a, 0)$ and $B(0, b)$ respectively. The coordinate of the point which divides the line joining $A(a, 0)$ and $B(0, b)$ in the ratio $3: 2$ are
$
\left(\frac{3 \times 0+2 \times a}{3+2}, \frac{3 \times b+2 \times 0}{3+2}\right) \Rightarrow\left(\frac{2 a}{5}, \frac{3 b}{5}\right)
$
If is given that, the point $(2,-1)$ divides the $A B$ in the ratio $3: 2$.
$
\therefore \frac{2 a}{5}=2 \text { and } \frac{3 b}{5}=-1 \Rightarrow a=5 \text { and } b=\frac{-5}{3}
$
Hence, the required equation of the required line is
$
\begin{array}{rlrl}
& & \frac{x}{5}+\frac{y}{\frac{-5}{3}} & =1 \\
\Rightarrow & \frac{x}{5}-\frac{3 y}{5} & =1 \\
\Rightarrow & x-3 y & =5 \\
\Rightarrow & x-3 y-5 & =0
\end{array}
$