# 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 $AB$.

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Given:

$A$ and $B$ are two points having coordinates $(-2, -2)$ and $(2, -4)$ respectively.

To do:

We have to find the coordinates of $P$ such that $AP = \frac{3}{7} AB$.

Solution:

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

$PB = (1-\frac{3}{7})$

$AB=\frac{4}{7}AB$.

This implies,

$AP:PB=\frac{3}{7}AB:\frac{4}{7}AB=3:4$

Point $P$ divides the line segment joining the points $A(-2, -2)$ and $B(2, -4)$ in the ratio of $3 : 4$.

Using section formula, we have,

$(x, y)=(\frac{mx_{2}+nx_{1}}{m+n}, \frac{my_{2}+ny_{1}}{m+n})$

Therefore,

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

$=(\frac{6-8}{7}, \frac{-12-8}{7})$

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

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

Updated on 10-Oct-2022 13:22:05