Question

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

Answer 1

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

As, $AP=\frac{3}{7}AB$, then,

$AP=\frac{3}{7}\left(AP+PB\right)$

$7AP=3AP+3PB$

$4AP=3PB$

$\frac{AP}{PB}=\frac{3}{4}$

So, the point P divides the line segment in the ratio $3:4$.

By section formula,

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

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

$x=\frac{-2}{7},y=\frac{-20}{7}$

So, the coordinates of P are $\left(-\frac{2}{7},-\frac{20}{7}\right)$.