For what value of p does the equation y^2 + x | vedclass

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(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^

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$\frac{1}{4}$

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(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 $px^2 + xy + y^2 - x - 2y + 0 = 0$ with the general form:$a = p, h = \frac{1}{2}, b = 1, g = -\frac{1}{2}, f = -1, c = 0$.Substituting these values into the condition:$p(1)(0) + 2(-1)(-\frac{1}{2})(\frac{1}{2}) - p(-1)^2 - 1(-\frac{1}{2})^2 - 0(\frac{1}{2})^2 = 0$.$0 + \frac{1}{2} - p - \frac{1}{4} = 0$.$-p + \frac{1}{4} = 0$.$p = \frac{1}{4}$.

For what value of $p$ does the equation $y^2 + xy + px^2 - x - 2y = 0$ represent two straight lines?

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