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Chapter 11: Problem 31

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

Short Answer

Expert verified
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

01

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: $$x = r\cos\theta$$ $$y = r\sin\theta$$
02

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: $$x^2 + y^2 = 2y + 2x$$
03

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: $$(x - 1)^2 + (y - 1)^2 = 2$$
04

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 \(\sqrt{2}\).

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Most popular questions from this chapter

Find an equation of the line tangent to the hyperbola \(x^{2} / a^{2}-y^{2} / b^{2}=1\) at the point \(\left(x_{0}, y_{0}\right)\)

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