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Q.
The line parallel to the x-axis and passing through the point of intersection of the lines $ ax+2by+3b=0 $ and $ bx-2ay-3a=0, $ where $ (a,b)\ne (0,0) $ is
Let the line parallel to $ x- $ axis is, $ y=c $ ...(i) Given lines, $ ax+2by=-3b $ ...(ii) $ bx-2ay=3a $ ...(iii) Multiply by b in Eq. (ii) and by a in Eq. (iii), then subtract Eq. (ii) from Eq. (iii), $ \begin{align} & \underline{\begin{align} & abx+2{{b}^{2}}y=-3{{b}^{2}} \\ & xab-2{{a}^{2}}y=3{{a}^{2}} \\ & -\,\,\,\,\,\,\,\,+\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,- \\ \end{align}} \\ & 2({{a}^{2}}+{{b}^{2}})y=-3({{a}^{2}}+{{b}^{2}}) \\ \end{align} $
$ \Rightarrow $ $ y=-\frac{3}{2} $ From Eq. (i), $ c=-\frac{3}{2} $ Hence, the line is, $ y=-\frac{3}{2} $ Which is passing through the point of intersection of the lines Eqs. (ii) and (iii) below the $ x- $ axis at a distance of $ \frac{3}{2} $ .