Question

If $A(1,1)$ and $B(2,-3)$ are two points and $P(h,k)$ lies on the line $AB$ such that $PA=3AB$, where $B$ lies in between $A$ and $P$, then the value of $h-k$ is

Answer 1

Given : $AP=3AB$
Let $AB=x$, we get
$AP=3x\Rightarrow BP=3x-x=2x\Rightarrow \frac{AP}{BP}=\frac{3}{2}$


Hence, $P$ divides $AB$ externally in the ratio $3:2.$
$\therefore P\equiv (\frac{3\times 2-2\times 1}{3-2},\frac{3\times (-3)-2\times 1}{3-2})$
$\Rightarrow P\equiv (4,-11)\Rightarrow h=4,k=-11\therefore h-k=15$