The combined equation of the pair of lines through the point $$\left( {1,\,0} \right)$$  and parallel to the lines represented by $$2{x^2} - xy - {y^2} = 0$$     is :

Solution :
We have the equation
$$\eqalign{ & 2{x^2} - xy - {y^2} = 0 \cr & \Rightarrow \left( {2x + y} \right)\left( {x - y} \right) = 0 \cr} $$
If $$\left( {h,\,k} \right)$$  be the point then remaining pair is $$\left( {2x + y + h} \right)\left( {x - y + k} \right) = 0$$
Where, $$2x + y + h = 0$$    and $$x - y + k = 0$$
It passes through the point $$\left( {1,\,0} \right)$$
$$\eqalign{ & \therefore \,2 \times 1 + 0 + h = 0 \Rightarrow 2 + h = 0 \Rightarrow h = - 2 \cr & {\text{and }}1 - 0 + k = 0 \Rightarrow 1 + k = 0 \Rightarrow k = - 1 \cr} $$
$$\therefore $$  Required pair is
$$\eqalign{ & \left( {2x + y - 2} \right)\left( {x - y - 1} \right) = 0 \cr & \Rightarrow 2{x^2} - 2xy - 2x + xy - {y^2} - y - 2x + 2y + 2 = 0 \cr & \therefore \,2{x^2} - xy - {y^2} - 4x + y + 2 = 0 \cr} $$