The value of lambda, for which the equation x | vedclass

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Question

(A) The given equation is $x^2 - y^2 - x - \lambda y - 2 = 0$. Comparing this with the general second-degree equation $ax^2 + 2hxy + by^2 + 2gx + 2fy

Vedclass · Accepted Answer

$3, -3$

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(A) The given equation is $x^2 - y^2 - x - \lambda y - 2 = 0$. Comparing this with the general second-degree equation $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$, we get: $a = 1, b = -1, c = -2, h = 0, g = -\frac{1}{2}, f = -\frac{\lambda}{2}$. For the equation to represent a pair of straight lines, the condition is $abc + 2fgh - af^2 - bg^2 - ch^2 = 0$. Substituting the values: $(1)(-1)(-2) + 2(-\frac{\lambda}{2})(-\frac{1}{2})(0) - (1)(-\frac{\lambda}{2})^2 - (-1)(-\frac{1}{2})^2 - (-2)(0)^2 = 0$. $2 + 0 - \frac{\lambda^2}{4} + \frac{1}{4} = 0$. $2 + \frac{1}{4} = \frac{\lambda^2}{4}$. $\frac{9}{4} = \frac{\lambda^2}{4}$. $\lambda^2 = 9 \Rightarrow \lambda = \pm 3$.

The value of $\lambda$, for which the equation $x^2 - y^2 - x - \lambda y - 2 = 0$ represents a pair of straight lines, is

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