# If A and B are two points having coordinates (−2, −2) and (2, −4) respectively,

Question:

If $A$ and $B$ are two points having coordinates $(-2,-2)$ and $(2,-4)$ respectively, find the coordinates of $P$ such that $A P=\frac{3}{7} A B$.

Solution:

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,

$\mathrm{P}(x, y)=\left(\frac{3(2)+4(-2)}{3+4}, \frac{3(-4)+4(-2)}{3+4}\right)$

$=\left(-\frac{2}{7},-\frac{20}{7}\right)$