Cross multiplication gives a second degree equation
( x + y - 2 )( y + 1 ) = -2(x+y−2)(y+1)=−2
that represents a hyperbola, with asymptotes
( x + y - 2 )( y + 1 ) = 0(x+y−2)(y+1)=0.
Using ( x, y ) = r ( cos theta, sin theta )(x,y)=r(cosθ,sinθ),
r sin theta = (r(sin theta - cos theta ))/(r ( sin theta + cos theta ))rsinθ=r(sinθ−cosθ)r(sinθ+cosθ)
giving
r sin theta = ( tan theta - 1 ) / ( tan theta + 1 )rsinθ=tanθ−1tanθ+1
= ( tan theta - tan (pi/4 ))/( 1 + tan theta tan (pi/4) )=tanθ−tan(π4)1+tanθtan(π4)
= tan ( theta - pi/4 )=tan(θ−π4)
See graph for the hyperbola and asymptotes
r = -csc thetar=−cscθ
and r = 2/( cos theta + sin theta ) = sqrt2csc ( theta + pi/4)r=2cosθ+sinθ=√2csc(θ+π4)
graph{ (y - (y-x)/(y+x))(y+1)(x+y-2)=0[-5 15 -6 4]}
