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.

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