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

R.E.F image

Given : two pouts (1,0) & (2,3)

point winch divined line segment joining above

two points in ratio 1:n :-

$P(x=1)=\frac{1\times 2+n\times 1}{n+1}=\frac{n+2}{n+1}$

$P(y=1)=\frac{1\times 3+n\times 0}{n+1}=\frac{3}{n+1}$

So, $P(x,y)\equiv (\frac{n+2}{n+1},\frac{3}{n+1})$

Slope of given line segment -

$m_1=\frac{y_2-y_1}{x_2-x_1}=\frac{3-0}{2-1}=3$

we know that slope of perpendicular line to given

line segment will be on-

$m_2=\frac{-1}{m_1}\Rightarrow m_2=\frac{-1}{3}$

Now for a line we have a point and slope, so

equation we can write on -

$y-y_1=m_2(x-x_1)\Rightarrow (y-\frac{3}{n+1})=\frac{-1}{3}(x-\frac{n+2}{n+1})$

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

$3(n+1)y-9=(n+2)-(n+1)x$

$3(n+1)y+(n+1)x=n+11\Rightarrow 3y+x=\frac{n+11}{n+1}$