Question

Sol. [Key Concept: Pair of lines parallel to a given pair through a point: translate the origin — replace $(x,y)$ by $(x - 3,\ y + 2)$ in the pair.]

$$(x - 3y)(x - y) = 0$$

Equation of line parallel to line $x - 3y = 0$ and passing through $(3, - 2)$ is

$L_{1} \equiv x - 3y = 9$

Similarly equation of line, parallel to line $x - y = 0$ and passing through $(3, - 2)$ is $L_{2} \equiv x - y = 5$

∴ Equation of pair $L_{1}$ and $L_{2}$ is $(x - 3y - 9)(x - y - 5) = 0$

$$\Rightarrow x^{2} - 4xy + 3y^{2} - 14x + 24y + 45 = 0$$