Question

The combined equation of the bisectors of the angle between the lines represented by $(x^2+y^2)\sqrt{3}=4xy$ is

• A. $y^2-x^2=0$
• B. $xy=0$
• C. $x^2+y^2=2xy$
• D. $\frac{x^2-y^2}{\sqrt{3}}=\frac{xy}{2}$

Answer 1

The correct option is A $y^2-x^2=0$

$\sqrt{3}{x}^2+\sqrt{3}{y}^2=4xy$ (Equation of given line)

or $\sqrt{3}{x}^2-4xy+\sqrt{3}{y}^2=0$

$\sqrt{3}{x}^2-3xy-xy+\sqrt{3}y=0$

$\sqrt{3}x(x-\sqrt{3}y)-y(x-\sqrt{3}y)=0$

$(x-\sqrt{3}y)(\sqrt{3}x-y)=0$ ($2$ lines)

$\therefore y=\frac{x}{\sqrt{3}},y=\sqrt{3}x$

$y=x\tan {30}^o,y=x\tan {60}^o$

$\therefore$ combined equation

$(y-x\tan 45)(y-x\tan 135)=0$

$(y-x)(y+x)=0$

$y^2-x^2=0$