Question

Find the ratio in which the $y-$axis divides the line segment joining the points $(-4,-6)$ and $(10,12)$. Also, find the coordinates of the point of division.

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 $y-$axis divides the line joining points $A(-4,-6)$ and $B(10,12)$ in ratio $y:1$

Then, as per section formula the coordinates of point which divides the line is $\frac{10y-4}{y+1},\frac{12y-6}{y+1}$

We know that coordinate at $y-$axis of point of $x$ is zero

Then, $\frac{10y-4}{y+1}=0$

$\Rightarrow 10y-4=0$

$\Rightarrow 10y=4$

$\Rightarrow y=\frac{10}{4}=\frac{5}{2}$

Then, ratio is $\frac{2}{5}:1\Rightarrow 2:5$

Substitute the value of $y$ in $y-$ coordinates, we get

$\frac{12\frac{2}{5}-6}{\frac{2}{5}+1}=\frac{24-30}{2-5}=\frac{-6}{-3}=2$

Then, coordinates of point which divides the line joining $A$ and $B$ is $(0,2)$ and ratio $\frac{2}{5}$.