An equation that expresses a figure in terms of (r, θ) is called a polar equation.
- Circle centered at the origin with radius a: r = a
- Line passing through the origin and making an angle α with the polar axis: θ = α
- Circle with center (a, 0) and radius a: r = 2a cosθ
Example ① What does r = 4cosθ represent?
Solution: Multiply both sides by r: r² = 4r cosθ → x²+y² = 4x → (x−2)²+y² = 4
A circle with center (2,0) and radius 2
Solution: Multiply both sides by r: r² = 4r cosθ → x²+y² = 4x → (x−2)²+y² = 4
A circle with center (2,0) and radius 2
Key Point
- Use r² = x²+y², r cosθ = x, and r sinθ = y for conversion.
Practice Problems
- What figure is represented by r = 4sinθ?
Solutions and Explanations
- Solution: A circle with center (0,2) and radius 2