Question

In the given figure $P(3,1)$ is a point on the line segment $AB$ that $AP:PB=2:3$. Find the co-ordinates of $A$ and $B$.

Answer 1

Using the section formula, if a point $(x,y)$ divides the line joining the points $(x_1,y_1)$ and $(x_2,y_2)$ in the ratio $m:n$, then

$(x,y)=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$

Let the coordinates of $A$ be $(a,0)$ and B be $(0,b)$

It is gien that $P$ divides segment $AB$. Thus by section's formula,

$3=\frac{3a+0}{5}$

$\Rightarrow a=5$

$1=\frac{0+2b}{5}$

Also, $b=\frac{5}{2}$

$\Rightarrow A(5,0)$

$B(0,\frac{5}{2})$.