# Current Affairs JEE Main & Advanced

## 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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