Question

In what ratio the line joining the point $(2,3)(4,1)$ divides the segment joining the point $(1,2)$ and $(4,3)$.

Answer 1

Let $P(2,3),Q(4,1),R(1,2),S(4,3)$ be the points

Equation of $PQ:\frac{y-1}{x-4}=\frac{2}{-2}\Rightarrow x+y=5$ ___ (1)

Equation of $RS:\frac{y-3}{x-4}=\frac{1}{3}\Rightarrow 3y-x=\left(3\right)-\left(2\right)$

Adding (1), (2), $4y=10\Rightarrow y=5/2,x=5/2$

$\therefore x(\frac{5}{2},\frac{5}{2})$ is the point of intersection

Let x divide RS in $m:1$ ratio, then

$\left(\frac{3m+2\left(1\right)}{m+1}\right)=\frac{5}{2}$ $[\because \frac{m{y}_2+n{y}_1}{m+n}=y]$

$6m+4=5m+5$

$m=1$

$\therefore$ It divides RS in $1:1$ ratio