JEE Main & Advanced Mathematics Pair of Straight Lines Angle Between the Pair of Lines

The angle between the lines represented by     

\[a{{x}^{2}}+2hxy+b{{y}^{2}}+2gx+2fy+c=0\] or \[a{{x}^{2}}+2hxy+b{{y}^{2}}=0\]

is given by \[\tan \theta =\left| \frac{2\sqrt{{{h}^{2}}-ab}}{a+b} \right|\,\,\,\,\Rightarrow \theta ={{\tan }^{-1}}\left| \frac{2\sqrt{{{h}^{2}}-ab}}{a+b} \right|\]

From the above formula it is clear, that

(i) The lines represented by \[a{{x}^{2}}+2hxy+b{{y}^{2}}+2gx+2fy+c=0\] are parallel iff \[{{h}^{2}}=ab\] and \[a{{f}^{2}}=b{{g}^{2}}\] or \[\frac{a}{h}=\frac{h}{b}=\frac{g}{f}\].

(ii) The lines represented by \[a{{x}^{2}}+2hxy+b{{y}^{2}}\] \[+2gx+2fy+c=0\] are perpendicular iff \[a+b=0\]

i.e.,  Coefficient of \[{{x}^{2}}+\] Coefficient of \[{{y}^{2}}=0\].

(iii) The lines are coincident, if \[{{g}^{2}}=ac\].