Question

A line perpendicular to the line segment joining the points $(1,0)$ and $(2,3)$ divides it in the ratio $1:n$. Find the equation of the line.

Answer 1

The slope of the line joining $A(1,0)$ and $B(2,3)$ is $\frac{3-0}{2-1}=3$ and the coordinates of the point dividing it in the ratio $1:n$ are $(\frac{n+2}{n+1},\frac{3}{n+1})$.

The slope of the line perpendicular to the line segment AB is $-\frac{1}{3}$.

Hence, the equation of the required line is

$y-\frac{3}{n+1}=-\frac{1}{3}(x-\frac{n+2}{n+1})$

$(n+1)x+3(n+1)y=n+11$