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

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Question

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

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(C) The general equation of the second degree $ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0$ 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 standard form: $a = \lambda, b = 2, c = 3, h = -5/2, g = 5/2, f = -7/2$. 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 $4$: $24\lambda + 175 - 49\lambda - 50 - 75 = 0$ $-25\lambda + 50 = 0$ $25\lambda = 50 \implies \lambda = 2$.

If the equation $\lambda x^{2} + 2y^{2} - 5xy + 5x - 7y + 3 = 0$ represents a pair of straight lines,then $\lambda = \dots$

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