Bisectors of the Angles Between the Lines
Category : JEE Main & Advanced
(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.
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