Question

For what values of $a$ 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 axis.

Answer 1

$2x-3y+6=0⟶\left(1\right)$

Intercepts wrt on axes are

$x-$ intercept $y=0$ in eq$\left(1\right)$

$2x=-6\Rightarrow x=-3$

$y-$ intercept $x=0$ in eq$\left(1\right)$

$-3y=-6$

$y=2$

Now intercepts of $ax+by+8=0$ should be $3$ and $-2$

$x-$ intercept $ax=-8$ $y-$ intercept

$x=3$ $by=-8$

$a\left(3\right)=-8$ $b(-2)=-8$

$a=-8/3$ $b=4$

$\therefore$ , $a=\frac{-8}{3},b=4$