Question

For what values of a a and b the intercepts cut off on the coordinate axes by the line ax + by + 8 = 0 are equal in length but opposite in signs to those cut off by the line 2x - 3y + 6 = 0 on the axes.

Answer 1

$\text{The equation of the given line is,}ax+by+8=0\Rightarrow -\frac{x}{\frac{8}{a}}-\frac{y}{\frac{8}{b}}=1\text{It cuts the axes at A}(\frac{-8}{a},0)andB(0,\frac{-8}{b})\text{The equation of the given line is,}2x-3y+6=0\Rightarrow \frac{-x}{3}+\frac{y}{2}=1\text{It cuts the axes at C(-3, 0) and D(0, 2).}\text{The intercepts of both the lines are opposite in sign}\Rightarrow (\frac{-8}{a},0)=-1(-3,0)and(0,\frac{-8}{b})=-(0,2)\Rightarrow \frac{-8}{a}=3and\frac{-8}{b}=-2\Rightarrow a=\frac{-8}{3}andb=4$