If the equation lambda x^2 + 2y^2 - 5xy + 5x | vedclass

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(B) The general equation of a second-degree curve $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ represents a pair of straight lines if $abc + 2fgh - af^2 -

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$2$

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(B) The general equation of a second-degree curve $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ represents a pair of straight lines if $abc + 2fgh - af^2 - bg^2 - ch^2 = 0$.Comparing the given equation $\lambda x^2 - 5xy + 2y^2 + 5x - 7y + 3 = 0$ with the general form,we have:$a = \lambda, b = 2, c = 3, h = -5/2, g = 5/2, f = -7/2$.Substituting these values into the condition:$\lambda(2)(3) + 2(-7/2)(5/2)(-5/2) - \lambda(-7/2)^2 - 2(5/2)^2 - 3(-5/2)^2 = 0$$6\lambda + 175/4 - 49\lambda/4 - 25/2 - 75/4 = 0$Multiplying by $4$ to clear the denominators:$24\lambda + 175 - 49\lambda - 50 - 75 = 0$$-25\lambda + 50 = 0$$25\lambda = 50 \Rightarrow \lambda = 2$.

If the equation $\lambda x^2 + 2y^2 - 5xy + 5x - 7y + 3 = 0$ represents two straight lines,then the value of $\lambda$ is:

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