The equation 2x^2 + 7xy + 3y^2 + 8x + 14y + l | vedclass

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(D) The general equation of a second-degree curve $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ represents a pair of straight lines if the determinant $\De

Vedclass · Accepted Answer

$8$

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(D) The general equation of a second-degree curve $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ represents a pair of straight lines if the determinant $\Delta = abc + 2fgh - af^2 - bg^2 - ch^2 = 0$.Comparing the given equation $2x^2 + 7xy + 3y^2 + 8x + 14y + \lambda = 0$ with the general form:$a = 2, h = 7/2, b = 3, g = 4, f = 7, c = \lambda$.Substituting these values into the condition $\Delta = 0$:$(2)(3)(\lambda) + 2(7)(4)(7/2) - 2(7)^2 - 3(4)^2 - \lambda(7/2)^2 = 0$$6\lambda + 196 - 98 - 48 - 49\lambda/4 = 0$$6\lambda - 12.25\lambda + 50 = 0$$-6.25\lambda = -50$$\lambda = 8$.

The equation $2x^2 + 7xy + 3y^2 + 8x + 14y + \lambda = 0$ represents a pair of straight lines when $\lambda = $

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