Question

Calculate the ratio in which the line joining $A(-4,2)$ and $B(3,6)$ is divided by point $P(x,3)$. Also find length of $AP$

Answer 1

Let $P(x,3)$ divide the line segment joining the points $A(-4,2)$ and $B(3,6)$ in the ratio $k:1$
$\therefore$ Coordinates of $P$ is $(\frac{m_1{x}_2+m_2{x}_1}{m_1+m_2},\frac{m_1{y}_2+m_2{y}_1}{m_1+m_2})=(\frac{3k-4}{k+1},\frac{6k+2}{k+1})$
But coordinate of $P$ is $(x,3)$
$\Rightarrow \frac{6k+2}{k+1}=3$
$6k+2=3k+3$
$3k=1\Rightarrow k=\frac{1}{3}$
$\therefore$ The required ratio is $\frac{1}{3}:1$ i.e., $1:3$ (internally)
$\therefore$ Coordinate of $P$ is $(\frac{-9}{4},3)$
Length of $AP=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$
$=\sqrt{{(-\frac{9}{4}+4)}^2+(3-2{)}^2}$
$=\sqrt{{\left(\frac{-9+16}{4}\right)}^2+(1{)}^2}=\sqrt{\frac{49}{16}+1}$
$=\sqrt{\frac{49+16}{16}}=\sqrt{\frac{65}{16}}=\frac{\sqrt{65}}{4}$