If the equation ax^{2} + by^{2} + cx + cy = 0 | vedclass

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Question

(B) The general equation of a second-degree curve $ax^{2} + 2hxy + by^{2} + 2gx + 2fy + k = 0$ represents a pair of lines if the determinant of the ma

Vedclass · Accepted Answer

$a+b=0$

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(B) The general equation of a second-degree curve $ax^{2} + 2hxy + by^{2} + 2gx + 2fy + k = 0$ represents a pair of lines if the determinant of the matrix $\begin{vmatrix} a & h & g \ h & b & f \ g & f & k \end{vmatrix} = 0$. Comparing the given equation $ax^{2} + 0xy + by^{2} + cx + cy + 0 = 0$ with the general form,we have $h = 0$,$g = c/2$,$f = c/2$,and $k = 0$. Substituting these values into the determinant condition: $\begin{vmatrix} a & 0 & c/2 \ 0 & b & c/2 \ c/2 & c/2 & 0 \end{vmatrix} = 0$. Expanding along the first row: $a(0 - c^{2}/4) - 0 + (c/2)(0 - bc/2) = 0$. $-ac^{2}/4 - bc^{2}/4 = 0$. Multiplying by $-4$: $ac^{2} + bc^{2} = 0$. $c^{2}(a + b) = 0$. Since $c 
eq 0$,we must have $a + b = 0$.

If the equation $ax^{2} + by^{2} + cx + cy = 0$,$c \neq 0$ represents a pair of lines,then

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