If the coordinates of points A and B are $( -2,\ -2)$ and $( 2,\ -4)$ respectively, find the coordinates of P such that \ $AP=\frac{3}{7} AB$, where P lies on the line segment AB.
Given: The coordinates of points A and B are $( -2,\ -2)$ and $( 2,\ -4)$.
To do: To find the co-ordinates of P such that $AP=\frac{3}{7}AB$, where P lies on the line segment $AB$.
Solution:
Here, $P( x,\ y)$ divides line segment $AB$, such that,
$AP=\frac{3}{7}AB$
$\frac{AP}{AB} =\frac{3}{7}$
$\frac{AB}{AP} =\frac{7}{3}$
On subtrating 1 from both sides,
$\frac{AB}{AP} -1=\frac{7}{3} -1$
$\Rightarrow \frac{AB-AP}{AP} =\frac{7-3}{3}$
$\Rightarrow \frac{BP}{AP} =\frac{4}{3}$
$\Rightarrow \frac{AP}{BP} =\frac{3}{7}$
$\Rightarrow$ AP:BP$=$3:7
Using formula,
We have$P( x,\ y) =( \frac{nx_{1} +mx_{2}}{m+n} ,\ \frac{ny_{1} +my_{2}}{m+n})$
The coordinates of $P( x,\ y) =( \frac{3\times 2+4( -2)}{3+4} ,\ \frac{3( -4) +4( -2)}{3+4})$
$Rightarrow x=\frac{-2}{7} \ \ and\ y=-\frac{20}{7}$
Therefore, The co-ordinates of $P( x,\ y) =( \frac{-2}{7} \ ,\ -\frac{20}{7})$.
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