The midpoint P of the line segment joining the points A(−10, 4) and B(−2, 0) lies on the line segment joining the points C(−9, −4) and D(−4, y).
Question:
The midpoint P of the line segment joining the points A(−10, 4) and B(−2, 0) lies on the line segment joining the points C(−9, −4) and D(−4, y). Find the ratio in which P divides CD. Also find the value of y.
Solution:
The midpoint of $A B$ is $\left(\frac{-10-2}{2}, \frac{4+0}{2}\right)=P(-6,2)$.
Let $k$ be the ratio in which $P$ divides $C D$. So
$(-6,2)=\left(\frac{k(-4)-9}{k+1}, \frac{k(y)-4}{k+1}\right)$
$\Rightarrow \frac{k(-4)-9}{k+1}=-6$ and $\frac{k(y)-4}{k+1}=2$
$\Rightarrow k=\frac{3}{2}$
Now, substituting $k=\frac{3}{2}$ in $\frac{k(\mathrm{y})-4}{k+1}=2$, we get
$\frac{y \times \frac{3}{2}-4}{\frac{3}{2}+1}=2$
$\Rightarrow \frac{3 y-8}{5}=2$
$\Rightarrow y=\frac{10+8}{3}=6$
Hence, the required ratio is 3 : 2 and y = 6.