If the combined equation of the pair of lines passing through $(1, 1)$ and parallel to lines represented by $x^{2} - 3 x y + y^{2} = 0$ is $x^{2} - 3 x y + y^{2} + α x + β y + γ = 0$ , then $α + β - γ =$

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Solution

$∵ x^{2} - 3 x y + y^{2} = 0$ passing through $(0, 0)$
So, the required equation can be obtained by shifting of the origin to $(1, 1)$
Hence $(x - 1)^{2} - 3 (x - 1) (y - 1) + (y - 1)^{2} = 0$
$\Rightarrow x^{2} + y^{2} - 3 x y + x + y - 1 = 0 ∴ α = 1, β = 1, γ = - 1 \Rightarrow α + β - γ = 3$

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