Question

If $A$ and $B$ are $(-2,-2)$ and $(2,-4)$, respectively, find the coordinates of $P$ such that $AP$ $=\frac{3}{7}AB$ and $P$ lies on the line segment $AB$.

Answer 1

As given the coordinates of $A(-2,-2)$ and $B(2,-4)$ and $P$ is a point lies on $AB$.

And $AP=\frac{3}{7}AB$

$\therefore BP=\frac{4}{7}$

Then, ratio of $AP$ and $PB=m_1:m_2=3:4$

Let the coordinates of $P$ be $(x,y)$.

$\therefore x=\frac{m_1{x}_2+m_2{x}_1}{m_1+m_2}$

$\Rightarrow x=\frac{3\times 2+4\times (-2)}{3+4}=\frac{6-8}{7}=\frac{-2}{7}$

And $y=\frac{m_1{y}_2+m_2{y}_1}{m_1+m_2}$

$\Rightarrow y=\frac{3\times (-4)+4\times (-2)}{3+4}=\frac{-12-8}{7}=\frac{-20}{7}$

$\therefore$Coordinates of $P=$$\frac{-2}{7},\frac{-20}{7}$