Question

Transform the polar equation r² = a²cos2θ into cartensian form.  

Answer 1

The given equation is 

         r² = a² cos2θ  

or      r² = a² (cos²θ - sin²θ) ,      ( so, cos2θ = cos²θ - sin²θ)

Putting cosθ = x/r, sinθ = y/r and r =√(X² + y²) , we get 

or       r²(x²+y²) = a²(x²_y₂)

or      (X²+y²)(X²+y²) = a²(x²-y²) 

or      (X²+y²) = a²(x²_y²)  

 This is the required equation in cartesian form.