Question

Find the ratio in which the line segment joining the points $P(x,2)$ divides the line segment joining the points $A(12,5)$ and $B(4,-3)$. Also find the value of $x$.

Answer 1

Given that, points $P,A,B$ are collinear

So slope of $PA=$ Slope of $AB$

$\Rightarrow$ $\frac{5-2}{12-x}=\frac{-3-5}{4-12}$

$\Rightarrow$$\frac{3}{12-x}=\frac{-8}{-8}=1$

$\Rightarrow$$3=12-x\Rightarrow x=9$

$\Rightarrow$ So point $P(9,2)$

Ratio of $\frac{AP}{PB}=\frac{\sqrt{(12-9{)}^2+(5-2{)}^2}}{\sqrt{(4-9{)}^2+(-3-2{)}^2}}=\frac{\sqrt{18}}{\sqrt{50}}=\frac{3\sqrt{2}}{5\sqrt{2}}=\frac{3}{5}$

The point $P$ divides AB in the ration $3:5$ internally