Question

If the point $P\left(\frac{1}{2},y\right)$ lies on the line segment joining the points A(3, −5) and B(−7, 9) then find the
ratio in which P divides AB. Also, find the value of y.

Answer 1

Let the point $P\left(\frac{1}{2},y\right)$ divides the line segment joining the points A(3, −5) and B(−7, 9) in the ratio k : 1. Then
$$\begin{gathered} \left(\frac{1}{2},y\right)=\left(\frac{k\left(-7\right)+3}{k+1},\frac{k\left(9\right)-5}{k+1}\right) \\ \Rightarrow \frac{-7k+3}{k+1}=\frac{1}{2}\mathrm{and}\frac{9k-5}{k+1}=y \\ \Rightarrow k+1=-14k+6\Rightarrow k=\frac{1}{3} \end{gathered}$$
Now, substituting $k=\frac{1}{3}$ in $\frac{9k-5}{k+1}=y$, we get
$\frac{{\displaystyle \frac{9}{3}}-5}{{\displaystyle \frac{1}{3}}+1}=y\Rightarrow y=\frac{9-15}{1+3}=-\frac{3}{2}$
Hence, required ratio is 1 : 3 and $y=-\frac{3}{2}$.