Question

Convert the following equations to Cartesian coordinates. Describe the resulting curve. $$r=2 \sin \theta+2 \cos \theta$$

Short answer

The Cartesian equation of the resulting curve is $$(x - 1)^2 + (y - 1)^2 = 2$$, and it represents a circle with its center at the point (1, 1) and a radius of \(\sqrt{2}\).

Step-by-step solution

Write down the polar equation and the conversion formulas

The polar equation is given as: $$r=2 \sin \theta+2 \cos \theta$$ And the conversion formulas are: $$x = r\cos\theta$$ $$y = r\sin\theta$$

Rewrite the polar equation using the conversion formulas

Multiply both sides of the polar equation by r to get: $$r^2 = r(2 \sin \theta+2 \cos \theta)$$ Now substitute x and y using the conversion formulas: $$x^2 + y^2 = 2y + 2x$$

Simplify the Cartesian equation

Rearrange the equation to get: $$x^2 - 2x + y^2 - 2y = 0$$ Now complete the square for both x and y: $$(x^2 - 2x + 1) + (y^2 - 2y + 1) = 1 + 1$$ This simplifies to: $$(x - 1)^2 + (y - 1)^2 = 2$$

Identify the resulting curve

The simplified equation, $$(x - 1)^2 + (y - 1)^2 = 2$$, represents a circle in Cartesian coordinates. The circle has its center at the point (1, 1) and a radius of \(\sqrt{2}\).