Geo3.5

Find the equation to the two lines represented by the equation $x^2-2xy\csc\theta+y^2=0\,$

Given $x^2-2xy\csc\theta+y^2=0\,$

$x=\frac{-2y\csc\theta\pm\sqrt{4y^2\csc^2\theta-4y^2}}{2}=\frac{1}{2}[-2y\csc\theta\pm2y\sqrt{\csc^2\theta-1}]\,$

$x=-y\csc\theta\pm y\cot\theta\,$

$x=-y(\csc\theta\pm \cot\theta)\,$

Therefore the two lines are $x+y(\csc\theta+\cot\theta)=0, x+y(\csc\theta-\cot\theta)=0\,$