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

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)