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)

Answer 2

The polar equation of the curve $r=-cot\theta cosec\theta$.

Step-by-step explanation:

Given: Polar equation of the form $r=f(\theta)$ for the curve represented by the cartesian equation $x=-y^2$.

Polar coordinates are defined in complex number as:

$x=rcos\theta\;,\;y=rsin\theta$

So, substituting the value in given equation:

$rcos\theta=(-rsin\theta)^2\\rcos\theta=-r^2sin^2\theta$

      $r=-\frac{cos\theta}{sin^2\theta}\\r=-cot\theta cosec\theta$                          $(As\;cot\theta=\frac{cos\theta}{sin\theta},\;cosec\theta=\frac{1}{sin\theta} )$

Hence, the polar equation of the curve of the form $r=f(\theta)$ represented by the cartesian equation $x=-y^2$ is  $r=-cot\theta cosec\theta$.

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