# In what ratio does the point

Question:

In what ratio does the point $\left(\frac{24}{11}, y\right)$ divide the line segment joining the points $\mathrm{P}(2,-2)$ and $\mathrm{Q}(3,7) ?$ Also, find the value of $y .$

Solution:

Let the point $\mathrm{P}\left(\frac{24}{11}, y\right)$ divides the line $\mathrm{PQ}$ in the ratio $k: 1$

Then, by the section formula:

$x=\frac{m x_{2}+n x_{1}}{m+n}, y=\frac{m y_{2}+n y_{1}}{m+n}$

The coordinates of $\mathrm{R}$ are $\left(\frac{24}{11}, y\right)$.

$\frac{24}{11}=\frac{3 k+2}{k+1}, y=\frac{7 k-2}{k+1}$

$\Rightarrow 24(k+1)=33 k+22, y(k+1)=7 k-2$

$\Rightarrow 24 k+24=33 k+22, \quad y k+y=7 k-2$

$\Rightarrow 2=9 k$

$\Rightarrow k=\frac{2}{9}$

Now consider the equation $y k+y=7 k-2$ and put $k=\frac{2}{9}$.

$\Rightarrow \frac{2}{9} y+y=\frac{14}{9}-2$

$\Rightarrow \frac{11}{9} y=\frac{-4}{9}$

$\Rightarrow y=\frac{-4}{11}$

Therefore, the point $\mathrm{R}$ divides the line $\mathrm{PQ}$ in the ratio $2: 9 .$

And, the coordinates of $R$ are $\left(\frac{24}{11}, \frac{-4}{11}\right)$.