Question

Find the ratio in which the point $P(-1,y)$ lying on the line segment joining $A(-3,10)$ and $B(6,-8)$ divides it. Also find the value of $y$.

Answer 1

Using the section formula, if a point $(x,y)$ divides the line joining the points $(x_1,y_1)$ and $(x_2,y_2)$ in the ratio $m:n$, then

$(x,y)=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$

Let the ratio be $m:n$

Then,$\frac{m{x}_2+n{x}_1}{m+n}=-1$

$\Rightarrow \frac{6m+3n}{m+n}=-1$

$\Rightarrow 6m-3n=-m-n$

$\Rightarrow 7m=2n$

$\Rightarrow \frac{m}{n}=\frac{2}{7}$

$\Rightarrow m:n=2:7$

Now, $\frac{m{y}_2+n{y}_1}{m+n}=y$

$\Rightarrow \frac{2(-8)+7\left(10\right)}{2+7}=y$

$\Rightarrow -16+70=9y$

$\Rightarrow 9y=54$

$\Rightarrow y=\frac{54}{9}=6$