Step 1 :
It is given that the intercepts cut off on the coordinate axes by the line $ ax+by+8=0$------(1) are equal in length but opposite in signs to those cut off by the line. $2x-3y+6=0$-------(2)
This implies that the slope of the two lines are equal.
Slope of line (1) is $ -\large\frac{a}{b}$
Slope of line (2) is $ \large\frac{2}{3}$
$ \therefore -\large\frac{a}{b} = \large\frac{2}{3}$
$ \Rightarrow a = \large\frac{-2b}{3}$
Step 2 :
The length of the perpendicular from the origin to the line (1) is
$ d_1 = \bigg| \large\frac{a(0)+b(0)+8}{\sqrt{a^2+b^2}} \bigg|$
Substituting for $a$ we get,
$ d_1 = \large\frac{8 \times 3 }{ \sqrt{13b^2}}$
Length of the perpendicular from the origin to line $(l_2)$ is
$ d_2 = \bigg| \large\frac{2(0)-3(0)+6}{\sqrt{2^2+3^2}} \bigg|$
But $d_1=d_2$
$ \therefore \large\frac{8 \times 3 }{\sqrt{13b^2}} $$ = \large\frac{6}{13}$
$ \Rightarrow b = 4$
$ \therefore a = \large\frac{-2b}{3}$$ = \large\frac{-2 \times 4}{3}$$ = \large\frac{-8}{3}$
hence the value of $a$ and $b$ are $ \large\frac{-8}{3}$$, 4$