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(C) The general equation of a second-degree curve is $Ax^{2} + 2Hxy + By^{2} + 2Gx + 2Fy + C = 0$. For this to represent a pair of lines,the condition

Vedclass · Accepted Answer

$0, \frac{15}{2}$

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(C) The general equation of a second-degree curve is $Ax^{2} + 2Hxy + By^{2} + 2Gx + 2Fy + C = 0$. For this to represent a pair of lines,the condition is $\Delta = ABC + 2FGH - AF^{2} - BG^{2} - CH^{2} = 0$. Comparing the given equation $kxy + 5x + 3y + 2 = 0$ with the general form,we have: $A = 0, B = 0, C = 2, 2H = k$ $\Rightarrow H = \frac{k}{2}, 2G = 5$ $\Rightarrow G = \frac{5}{2}, 2F = 3$ $\Rightarrow F = \frac{3}{2}$. Substituting these values into the condition $\Delta = 0$: $0 + 2(\frac{3}{2})(\frac{5}{2})(\frac{k}{2}) - 0 - 0 - 2(\frac{k}{2})^{2} = 0$. $\frac{15k}{4} - \frac{2k^{2}}{4} = 0$. $15k - 2k^{2} = 0$. $k(15 - 2k) = 0$. Thus,$k = 0$ or $k = \frac{15}{2}$.

If the equation $kxy + 5x + 3y + 2 = 0$ represents a pair of lines,then $k=$

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