Question

Convert the following equations to Cartesian coordinates. Describe the resulting curve. $$r=2$$

Short answer

Answer: The Cartesian coordinate equation for the polar coordinate equation r = 2 is x^2 + y^2 = 4. This equation represents a circle with a center at the origin (0, 0) and a radius of 2.

Step-by-step solution

Use conversion formulas to convert the polar equation to Cartesian coordinates

Since the given equation is r = 2, we will replace r with 2 in the conversion formulas: x = 2 * cos(θ) y = 2 * sin(θ)

Eliminate the θ variable to obtain a Cartesian equation.

To describe the resulting curve, we need a single Cartesian equation without θ. We can achieve this by using the Pythagorean identity: sin^2(θ) + cos^2(θ) = 1. We start by finding formulas for sin^2(θ) and cos^2(θ) from our previous step: sin^2(θ) = (y/2)^2 cos^2(θ) = (x/2)^2 Now we can plug these formulas into the Pythagorean identity: (y/2)^2 + (x/2)^2 = 1

Simplify the resulting equation and describe the curve

We start by simplifying the equation: y^2/4 + x^2/4 = 1 To simplify further, we can multiply both sides by 4: y^2 + x^2 = 4 This is the equation of a circle in Cartesian coordinates with a center at the origin (0, 0) and a radius of 2.