Locus of the image of the point $\left(1,2\right)$ in the line $\left(x-y+2\right)+\lambda \left(2x-3y+1\right)=0$ where $\lambda$ is parameter is

Find the equations to the straight lines passing through the point of intersection of the straight lines $\mathrm{A}\mathrm{x}+\mathrm{B}\mathrm{y}+\mathrm{C}=0$ and ${\mathrm{A}}^{\text{'}}\mathrm{x}+{\mathrm{B}}^{\text{'}}\mathrm{y}+{\mathrm{C}}^{\text{'}}=0$
and

$\left(i\right)$  Passing through the origin.

$\left(ii\right)$ Parallel to the axis of $y$.

$\left(\mathrm{i}\mathrm{i}\mathrm{i}\right)$ Cutting off a given distance $a$ from the axis of $y$ and

$\left(iv\right)$ Passing through the given point $\left(x^{\text{'}},y^{\text{'}}\right)$.

The straight lines$u_1 = 0, u_2 = 0, u_3 = 0$  and $u_4= 0$Represent sides of a parallelogram as shown in adjacent figure, then-