Answer
$$2{\left( {x - \frac{1}{2}} \right)^2} + 2{\left( {y - \frac{1}{2}} \right)^2} = 1$$
Work Step by Step
$$\eqalign{
& r = \sin \theta + \cos \theta \cr
& {\text{Multiply both sides by }}r \cr
& {r^2} = r\sin \theta + r\cos \theta \cr
& {\text{Where }}{r^2} = {x^2} + {y^2},{\text{ }}r\sin \theta = y{\text{ and }}r\cos \theta = x \cr
& {x^2} + {y^2} = y + x \cr
& {x^2} - x + {y^2} - y = 0 \cr
& {\text{Complete the square}} \cr
& {x^2} - x + \frac{1}{4} + {y^2} - y + \frac{1}{4} = \frac{1}{2} \cr
& {\left( {x - \frac{1}{2}} \right)^2} + {\left( {y - \frac{1}{2}} \right)^2} = \frac{1}{2} \cr
& 2{\left( {x - \frac{1}{2}} \right)^2} + 2{\left( {y - \frac{1}{2}} \right)^2} = 1 \cr} $$
