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.

Answer 1

The midpoint of AB is $\left(\frac{-10-2}{2},\frac{4+0}{2}\right)=P\left(-6,2\right)$.
Let k be the ratio in which P divides CD. So
$$\begin{gathered} \left(-6,2\right)=\left(\frac{k\left(-4\right)-9}{k+1},\frac{k\left(y\right)-4}{k+1}\right) \\ \Rightarrow \frac{k\left(-4\right)-9}{k+1}=-6\mathrm{and}\frac{k\left(\mathrm{y}\right)-4}{k+1}=2 \\ \Rightarrow k=\frac{3}{2} \end{gathered}$$
Now, substituting $k=\frac{3}{2}$ in $\frac{k\left(\mathrm{y}\right)-4}{k+1}=2$, we get
$$\begin{gathered} \frac{y\times {\displaystyle \frac{3}{2}}-4}{{\displaystyle \frac{3}{2}}+1}=2 \\ \Rightarrow \frac{3y-8}{{\displaystyle 5}}=2 \\ \Rightarrow y=\frac{10+8}{3}=6 \end{gathered}$$
Hence, the required ratio is 3 : 2 and y = 6.