Question

What the is the polar form of `y = (y-x)/(x+y)`?

Answer 1

`r = tan ( theta - pi/4 ) csc theta`

Explanation:

Cross multiplication gives a second degree equation

`( x + y - 2 )( y + 1 ) = -2`

that represents a hyperbola, with asymptotes

`( x + y - 2 )( y + 1 ) = 0`.

Using `( x, y ) = r ( cos theta, sin theta )`,

`r sin theta = (r(sin theta - cos theta ))/(r ( sin theta + cos theta ))`

giving

`r sin theta = ( tan theta - 1 ) / ( tan theta + 1 )`

`= ( tan theta - tan (pi/4 ))/( 1 + tan theta tan (pi/4) )`

`= tan ( theta - pi/4 )`

See graph for the hyperbola and asymptotes

`r = -csc theta`

and `r = 2/( cos theta + sin theta ) = sqrt2csc ( theta + pi/4)`
graph{ (y - (y-x)/(y+x))(y+1)(x+y-2)=0[-5 15 -6 4]}