If $A$ and $B$ are two points having coordinates $(-2,-2)$ and $(2,-4)$ respectively, find the coordinates of $P$ such that $A P=\frac{3}{7} A B$.
We have two points A (−2,−2) and B (2,−4). Let P be any point which divide AB as,
$\mathrm{AP}=\frac{3}{7} \mathrm{AB}$
Since,
$A B=(A P+B P)$
So,
$7 \mathrm{AP}=3 \mathrm{AB}$
$7 \mathrm{AP}=3(\mathrm{AP}+\mathrm{BP})$
$4 \mathrm{AP}=3 \mathrm{BP}$
$\frac{\mathrm{AP}}{\mathrm{BP}}=\frac{3}{4}$
Now according to the section formula if any point $P$ divides a line segment joining $A\left(x_{1}, y_{1}\right)$ and $B\left(x_{2}, y_{2}\right)$ in the ratio $m$ : $n$ internally than,
$\mathrm{P}(x, y)=\left(\frac{n x_{1}+m x_{2}}{m+n}, \frac{n y_{1}+m y_{2}}{m+n}\right)$
Therefore P divides AB in the ratio 3: 4. So,
$\mathrm{P}(x, y)=\left(\frac{3(2)+4(-2)}{3+4}, \frac{3(-4)+4(-2)}{3+4}\right)$
$=\left(-\frac{2}{7},-\frac{20}{7}\right)$
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.