The combined equation of pair of lines passing through the point $(1, 2)$ and parallel to the pair of lines $x^{2} + 2 xy + y^{2} = 0$

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$x^{2} + y^{2} - 2 xy - 6 x - 6 y + 9 = 0$

$x^{2} + y^{2} + 2 xy - 6 x - 6 y - 9 = 0$

$x^{2} + y^{2} - 2 xy + 6 x + 6 y - 9 = 0$

$x^{2} + y^{2} + 2 xy - 6 x - 6 y + 9 = 0$

answer is D.

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Detailed Solution

The combined equation of pair of lines passing through the point $(x_{1}, y_{1})$ and parallel to the
pair of lines ${ax}^{2} + 2 hxy + {by}^{2} = 0$ is $a {(x - x_{1})}^{2} + 2 h (x - x_{1}) (y - y_{1}) + b {(y - y_{1})}^{2} = 0$
Hence, the required equation is ${(x - 1)}^{2} + 2 (x - 1) (y - 2) + {(y - 2)}^{2} = 0$
Simplify
$\begin{matrix} x^{2} - 2 x + 1 + 2 xy - 2 y - 4 x + 4 + y^{2} - 4 y + 4 = 0 \\ x^{2} + 2 xy + y^{2} - 6 x - 6 y + 9 = 0 \end{matrix}$
Therefore, the combined equation of pair of lines passing through the point (1,2) and parallel to the pair of lines $x^{2} + 2 x y + y^{2} = 0$ ,is $x^{2} + 2 xy + y^{2} - 6 x - 6 y + 9 = 0$ .