Question

The line parallel to the $x$-axis and passing through the intersection of the lines $ax+2by+3b=0$ and $bx-2ay-3a=0$ is, ( where $(a,b)\ne (0,0)$ )

Answer 1

$Equationofalinepassingthroughtheintersectionoflines(ax+2by+3b=0)and(bx-2ay-3a=0)is((ax+2by+3b)+n(bx-2ay-3a)=0)...\left(i\right)Now,thislineisparalleltox-axissocofficientofx=0=>(a+nb=0=>n=-\frac{a}{b})Onputtingthisvalueinequation\left(i\right)wegetb(ax+2by+3b)-a(bx-2ay-3a)=0=>2b^{2}y+3b^2+2a^{2}y+3a^2=0=>2(b^2+a^2)y+3(a^2+b^2)=0=>y=-\frac{3}{2}.$