Which among the following represents the comb | vedclass

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(A) The given equation is $2x^2 - xy - y^2 = 0$.Factoring the quadratic expression:$2x^2 - 2xy + xy - y^2 = 0$$2x(x - y) + y(x - y) = 0$$(2x + y)(x -

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$2x^2 - xy - y^2 - 4x + y + 2 = 0$

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(A) The given equation is $2x^2 - xy - y^2 = 0$.Factoring the quadratic expression:$2x^2 - 2xy + xy - y^2 = 0$$2x(x - y) + y(x - y) = 0$$(2x + y)(x - y) = 0$.The lines are parallel to $2x + y = 0$ and $x - y = 0$.Let the required lines be $(2x + y + k_1) = 0$ and $(x - y + k_2) = 0$.Since these lines pass through $(1, 0)$,we substitute the point:For the first line: $2(1) + 0 + k_1 = 0 \Rightarrow k_1 = -2$.For the second line: $1 - 0 + k_2 = 0 \Rightarrow k_2 = -1$.The combined equation is $(2x + y - 2)(x - y - 1) = 0$.Expanding the product:$2x(x - y - 1) + y(x - y - 1) - 2(x - y - 1) = 0$$2x^2 - 2xy - 2x + xy - y^2 - y - 2x + 2y + 2 = 0$$2x^2 - xy - y^2 - 4x + y + 2 = 0$.

Which among the following represents the combined equation of a pair of lines passing through the point $(1, 0)$ and parallel to the lines represented by $2x^2 - xy - y^2 = 0$?

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