Question

Find the ratio in which the point P(x, 2) divides the line segment joining the points A(12, 5) and B(4, −3). Also, find the value of x. [CBSE 2014]

Answer 1

Suppose P(x, 2) divides the line segment joining the points A(12, 5) and B(4, −3) in the ratio k : 1.

Using section formula, we get

Coordinates of P = $\left(\frac{4k+12}{k+1},\frac{-3k+5}{k+1}\right)$

$\therefore \left(\frac{4k+12}{k+1},\frac{-3k+5}{k+1}\right)=\left(x,2\right)$

$\Rightarrow x=\frac{4k+12}{k+1}$ and $\frac{-3k+5}{k+1}=2$

Now,

$$\begin{gathered} \frac{-3k+5}{k+1}=2 \\ \Rightarrow -3k+5=2k+2 \\ \Rightarrow 5k=3 \\ \Rightarrow k=\frac{3}{5} \end{gathered}$$

So, P divides the line segment AB in the ratio 3 : 5.

Putting k = $\frac{3}{5}$ in $x=\frac{4k+12}{k+1}$, we get
$x=\frac{4\times {\displaystyle \frac{3}{5}}+12}{{\displaystyle \frac{3}{5}}+1}=\frac{12+60}{3+5}=\frac{72}{8}=9$

Thus, the value of x is 9.