The line cuts $X$ and $Y$ axes at the points $A$ and $B$ respectively. The point $(5,6)$ divides the line…
The line cuts $X$ and $Y$ axes at the points $A$ and $B$ respectively. The point $(5,6)$ divides
the line segment $\mathrm{AB}$ internally in the ratio $3: 1$, then equation of line is
$2 x+y=16$
$2 x+5 y=40$
$2 x-y=4$
$2 x-5 y=-20$
Solution
Let $A \equiv(a, 0)$ and $B \equiv(0, b)$
Let $P \equiv(5,6)$ and it divides $A B$ in the ratio $3: 1$
$\begin{array}{l}
5=\frac{3 \times 0+1(a)}{3+1} \Rightarrow 5=\frac{a}{4} \Rightarrow a=20 \\
6=\frac{3 b+1 \times 0}{3+1} \Rightarrow 3 b=24 \Rightarrow b=8
\end{array}$
Thus intercepts on $X$ and $Y$ axes are 20 and 8 respectively. Equation of $\mathrm{AB}$ is $\frac{\mathrm{x}}{20}+\frac{\mathrm{y}}{8}=1$ i.e. $2 \mathrm{x}+5 \mathrm{y}=40$
