Question

Find the ratio in which the line $x-y=2$ divides the line segment joining the points $A(3,-1)$ and $B(8,9)$

Answer 1

Given equation line$=x-y=2$

$\Rightarrow y=x-2$

Equation of line AB is $(y-9)=\left(\frac{9-(-1)}{8-3}\right)(x-8)$

$8(y-9)=10(x-8)$

$y=2x-7$

$\therefore$Let intersection point of $y=x-2$ and $y=2x-7$ be P

$x-2=2x-7$

$x=5$

$\therefore y=3$

$\therefore P=(5,3)$

$AP=\sqrt{{(5-3)}^2+{(3+1)}^2}=2\sqrt{5}$

$BP=\sqrt{{(8-5)}^2+{(9-3)}^2}=3\sqrt{5}$

$\therefore AP:BP=2:3,\therefore P$ divides segment joining A and B in ratio $2:3$