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Q.
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
Straight Lines
Solution:
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 $(1, 0)$
$\therefore 2 × 1 + 0 + h = 0 \Rightarrow 2 + h = 0 \Rightarrow h=-2$
and $1-0+k=0 \Rightarrow 1+k=0 \Rightarrow k=-1$
$\therefore $ Required pair is $\left(2x + y - 2\right) \left(x - y - 1\right) = 0$
$\Rightarrow 2x^{2}-2xy+2x+xy-y^{2}-y-2x+2y+2=0$
$\therefore 2x^{2}-xy-y^{2}-4x+y+2=0$