Question

The line segment joining the points $(3,-4)$ and $(1,2)$ is trisected at the point $P$ and $Q.$ If the coordinates of $P$ and $Q$ are $(p,-2)$ and $(\frac{5}{3},q)$ respectively, then the value of $3p-q$ is

Answer 1

Let $A=(3,-4)$ and $B=(1,2)$
As line segment $AB$ is trisected, so line segment is either divided in ratio of $1:2$ or $2:1$

When $AB$ is divided in ratio of $1:2$, the coordinates
$=(\frac{1\left(1\right)+2\left(3\right)}{3},\frac{1\left(2\right)+2(-4)}{3})=(\frac{7}{3},-2)$

When $AB$ is divided in ratio of $2:1$, the coordinates
$=(\frac{2\left(1\right)+1\left(3\right)}{3},\frac{2\left(2\right)+1(-4)}{3})=(\frac{5}{3},0)$

Given coordinates of $P$ and $Q$ are $(p,-2)$ and $(\frac{5}{3},q)$ respectively, so
$p=\frac{7}{3},q=0\Rightarrow 3p-q=7$