Question
Show that the line segment joining the points $(-3, 10)$ and $(6, -5)$ is trisected by the coordinates axis.
✓
Answer
Let the coordinates of two points $x$-axis and $y$-axis be $P(x, O)$ and $G(0, y)$ respectively.
Let $P$ divides $A B$ in the ratio $k: 1$.
Coordinates of $P$ are
$ P(x, 0)=P\left(\frac{6 k-3}{k+1}, \frac{-5 k+10}{k+1}\right)$
$\Rightarrow 0=\frac{-5 k+10}{k+1}$
$\Rightarrow 5 k=10$
$\Rightarrow k=2 $
Hence $P$ divides $A B$ in the ratio $2: 1 .$
Let $Q$ divides $A B$ in the ratio $k_1: 1$.
Coordinates of $Q$ are,
$ Q (0, y )= Q \left(\frac{6 k _1-3}{ k +1}, \frac{-5 k +10}{ k +1}\right)$
$\Rightarrow 0=\frac{6 k _1-3}{ k +1}$
$\Rightarrow 6 k _1=3$
$\Rightarrow k _1=\frac{1}{2} $
Hence $Q$ divides $A B$ in the ratio $1:2$
Hence proved, $P$ and $Q $are the points of trisection.
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