Question

find a polar equation for the curve represented by the given cartesian equation. (assume 0 ≤ θ < 2π.)

x² + y² = 8y

Explanation:

Step1: Recall polar - Cartesian conversion formulas

We know that \(x = r\cos\theta\) and \(y = r\sin\theta\), and \(x^{2}+y^{2}=r^{2}\).

Step2: Substitute into the given Cartesian equation

Given \(x^{2}+y^{2}=8y\). Substitute \(x = r\cos\theta\), \(y = r\sin\theta\) and \(x^{2}+y^{2}=r^{2}\) into the equation.
We get \(r^{2}=8r\sin\theta\).

Step3: Simplify the equation

Since \(r
eq0\) (if \(r = 0\), it represents the origin which is also included in the non - zero solutions of the simplified equation). Divide both sides of the equation \(r^{2}=8r\sin\theta\) by \(r\) (for \(r
eq0\)).
We obtain \(r = 8\sin\theta\).

Answer:

\(r = 8\sin\theta\)