Question

Find a polar equation of the form r=f(θ) for the curve represented by the cartesian equation x=−y2.

Answer 1

We define the following variables:
 x = r * cos (θ)
 y = r * sine (θ)
 Substituting the variables we have:
 x = -y ^ 2
 r * cos (θ) = - (r * sin (θ)) ^ 2
 Rewriting:
 r * cos (θ) = - (r ^ 2 * sin ^ 2 (θ))
 We cleared r:
 r = - ((cos (θ)) / (sin ^ 2 (θ)))
 We rewrite:
 r = - ((cos (θ)) / (sin (θ))) * (1 / sin (θ))
 r = - cot (θ) * csc (θ)
 Answer:
 a polar equation of the form r = f (θ) for the curve represented by the cartesian equation x = -y2 is:
 r = - cot (θ) * csc (θ)