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Polar Equations

Cartesian Equations (x² + y²)² = a² (x² - y²)

y=r sinθ & x=r cosθ

(r² cos²θ + r² sin²θ)² = a² (x² - y²)
(r² (cos²θ + sin²θ))²= a² (x² - y²)
     Pythagorean Identity
      cos²θ + sin²θ = 1
(r² (1) )² = a² (x² - y²)
(r²)² = a² (x² - y²)
r⁴ = a² (x² - y²)
 
r⁴ = a² (r² cos²θ - r²sin²θ)
r⁴ = a² (r² (cos²θ - sin²θ))
     Double Angle Formula
      cos²θ - sin²θ = cos(2θ)
r⁴ = a² r² cos(2θ)
r² = a² cos(2θ)