JEE Main & Advanced Mathematics Pair of Straight Lines Bisectors of the Angles Between the Lines

(1) The joint equation of the bisectors of the angles between the lines represented by the equation \[a{{x}^{2}}+2hxy+b{{y}^{2}}=0\] is

\[\frac{{{x}^{2}}-{{y}^{2}}}{a-b}=\frac{xy}{h}\Rightarrow h{{x}^{2}}-(a-b)xy-h{{y}^{2}}=0\]

Here, coefficient of \[{{x}^{2}}+\] coefficient of \[{{y}^{2}}=0\]. Hence, the bisectors of the angles between the lines are perpendicular to each other. The bisector lines will pass through origin also.

(i) If \[a=b\], the bisectors are \[{{x}^{2}}-{{y}^{2}}=0\].

i.e., \[x-y=0,x+y=0\]

(ii) If \[h=0\], the bisectors are \[xy=0\] i.e., \[x=0,y=0\].

(2) The equation of the bisectors of the angles between the lines represented by  \[a{{x}^{2}}+2hxy+b{{y}^{2}}+2gx+2fy+c=0\] are given by \[\frac{{{(x-\alpha )}^{2}}-{{(y-\beta )}^{2}}}{a-b}=\frac{(x-\alpha )(y-\beta )}{h}\], where \[\alpha ,\,\,\beta \] is the point of intersection of the lines represented by the given equation.