Question
Find the ratio in which the point $\text{P}\Big(\frac{3}{4},\frac{5}{12}\Big)$ divides the line segment joining the points $\text{A}\Big(\frac{1}{2},\frac{3}{2}\Big)$ and B(2, -5).
✓
Answer
Let the required ratio be k : 1.
Then, by section formula,
Coordinates of P $=\bigg(\frac{\text{k}\times2+1\times\frac{1}{2}}{\text{k}+1},\frac{\text{k}\times(-5)+1\times\frac{3}{2}}{\text{k}+1}\bigg)$
$=\bigg(\frac{2\text{k}+\frac{1}{2}}{\text{k}+1},\frac{-5\text{k}+\frac{3}{2}}{\text{k}+1}\bigg)$
$=\bigg(\frac{4\text{k}+1}{2(\text{k}+1)},\frac{-10\text{k}+3}{2(\text{k}+1)}\bigg)$
Given, coordinates of P $=\Big(\frac{3}{4},\frac{5}{12}\Big)$
$\therefore\ \frac{4\text{k}+1}{2(\text{k}+1)}=\frac{3}{4}$
$\Rightarrow\ 16\text{k}+4=6\text{k}+6$
$\Rightarrow\ 10\text{k}=2$
$\Rightarrow\ \text{k}=\frac{1}{5}$
So, the required ratio is 1 : 5.
Then, by section formula,
Coordinates of P $=\bigg(\frac{\text{k}\times2+1\times\frac{1}{2}}{\text{k}+1},\frac{\text{k}\times(-5)+1\times\frac{3}{2}}{\text{k}+1}\bigg)$
$=\bigg(\frac{2\text{k}+\frac{1}{2}}{\text{k}+1},\frac{-5\text{k}+\frac{3}{2}}{\text{k}+1}\bigg)$
$=\bigg(\frac{4\text{k}+1}{2(\text{k}+1)},\frac{-10\text{k}+3}{2(\text{k}+1)}\bigg)$
Given, coordinates of P $=\Big(\frac{3}{4},\frac{5}{12}\Big)$
$\therefore\ \frac{4\text{k}+1}{2(\text{k}+1)}=\frac{3}{4}$
$\Rightarrow\ 16\text{k}+4=6\text{k}+6$
$\Rightarrow\ 10\text{k}=2$
$\Rightarrow\ \text{k}=\frac{1}{5}$
So, the required ratio is 1 : 5.
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