If the equation 3x^2 + xy - y^2 - 3x + 6y + k | 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 lines if the determinant $\Delta = \be

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$-9$

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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 lines if the determinant $\Delta = \begin{vmatrix} a & h & g \ h & b & f \ g & f & c \end{vmatrix} = 0$.Comparing the given equation $3x^2 + xy - y^2 - 3x + 6y + k = 0$ with the general form,we have $a=3, h=1/2, b=-1, g=-3/2, f=3, c=k$.The condition for a pair of lines is $\begin{vmatrix} 3 & 1/2 & -3/2 \ 1/2 & -1 & 3 \ -3/2 & 3 & k \end{vmatrix} = 0$.Multiplying rows by $2$ to simplify the determinant: $\frac{1}{4} \begin{vmatrix} 6 & 1 & -3 \ 1 & -2 & 6 \ -3 & 6 & 2k \end{vmatrix} = 0$.Expanding the determinant: $6(-4k - 36) - 1(2k + 18) - 3(6 - 6) = 0$.$-24k - 216 - 2k - 18 = 0$.$-26k - 234 = 0$.$26k = -234$.$k = -9$.

If the equation $3x^2 + xy - y^2 - 3x + 6y + k = 0$ represents a pair of lines,then $k$ is equal to

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