Question

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

• A. $2x^2-xy-y^2-4x-y=0$
• B. $2x^2-xy-y^2-4x+y+2=0$
• C. $2x^2+xy+y^2-2x+y=0$
• D. None of these

Answer 1

Answer: (B)

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