Find the ratio in which the point P(x, 2) divides the line segmen… - Vidyadip

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 $x$.

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Answer

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 $\text{P}=\Big(\frac{4\text{k}+12}{\text{k}+1},\frac{-3\text{k}+5}{\text{k}+1}\Big)$
$\therefore\ \Big(\frac{4\text{k}+12}{\text{k}+1},\frac{-3\text{k}+5}{\text{k}+1}\Big)=(\text{x},2)$
$\Rightarrow\ \text{x}=\frac{4\text{k}+12}{\text{k}+1}$ and $\frac{-3\text{k}+5}{\text{k}+1}=2$
Now,
$\frac{-3\text{k}+5}{\text{k}+1}=2$
$\Rightarrow\ -3\text{k}+5=2\text{k}+2$
$\Rightarrow\ 5\text{k}=3$
$\Rightarrow\ \text{k}=\frac{3}{5}$
So, P divides the line segment AB in the ratio $3 : 5.$
Putting $\text{k}=\frac{3}{5}$ in $\text{x}=\frac{4\text{k}+12}{\text{k}+1},$ we get
$\text{x}=\frac{4\times\frac{3}{5}+12}{\frac{3}{5}+1}=\frac{12+60}{3+5}$
$=\frac{72}{8}=9$
Thus, the value of x is $9.$

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