(1) Equation of a pair of straight lines passing through origin : The equation \[a{{x}^{2}}+2hxy+b{{y}^{2}}=0\] represents a pair of straight line passing through the origin where \[a,\,\,h,\,\,b\] are constants.
Let the lines represented by \[a{{x}^{2}}+2hxy+b{{y}^{2}}=0\] be \[y-{{m}_{1}}x=0,\,\,y-{{m}_{2}}x=0\]. Then, \[{{m}_{1}}+{{m}_{2}}=-\frac{2h}{b}\] and \[{{m}_{1}}{{m}_{2}}=\frac{a}{b}\]
Then, two straight lines represented by \[a{{x}^{2}}+2hxy+b{{y}^{2}}=0\] are \[ax+hy+y\sqrt{{{h}^{2}}-ab}=0\] and \[ax+hy-y\sqrt{{{h}^{2}}-ab}=0\].
Hence, (a) The lines are real and distinct, if \[{{h}^{2}}-ab>0\]
(b) The lines are real and coincident, if \[{{h}^{2}}-ab=0\]
(c) The lines are imaginary, if \[{{h}^{2}}-ab<0\]
(2) General equation of a pair of straight lines : An equation of the form, \[a{{x}^{2}}+2hxy+b{{y}^{2}}+2gx+2fy+c=0\] where \[a,\,\,b,\,\,c,\,\,f,\,\,g,\,\,h\] are constants, is said to be a general equation of second degree in \[x\] and \[y\].
The necessary and sufficient condition for \[a{{x}^{2}}+2hxy+b{{y}^{2}}+2gx+2fy+c=0\] to represents a pair of straight lines is that \[abc+2fgh-a{{f}^{2}}-b{{g}^{2}}-c{{h}^{2}}=0\] or \[\left| \begin{matrix} a & h & g \\
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\[a{{x}^{2}}+2hxy+b{{y}^{2}}+2(gx+fy)\left( \frac{lx+my}{-n} \right)+c\,{{\left( \frac{lx+my}{-n} \right)}^{2}}=0\]
