To convert the rectangular equation $x^2 + y^2 - 2y = 0$ into polar form, we follow these steps:

Step 1: Recall the polar-to-rectangular relationships.

• $x = r \cos \theta$
• $y = r \sin \theta$
• $x^2 + y^2 = r^2$

Step 2: Substitute into the equation.

Start with the given equation: $x^2 + y^2 - 2y = 0$ Substitute $x^2 + y^2 = r^2$ and $y = r \sin \theta$ into the equation: $r^2 - 2(r \sin \theta) = 0$

Step 3: Simplify the equation.

Factor out $r$: $r(r - 2 \sin \theta) = 0$

Now, for this equation to be true, either:

• $r = 0$, which corresponds to the origin, or
• $r = 2 \sin \theta$, which is the desired polar form.

Thus, the polar form of the equation is: $r = 2 \sin \theta$

This represents a circle with a radius 1 and its center at $(0, 1)$ in the polar coordinate system.

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Would you like further clarification or a breakdown of any of the steps?

Here are some related questions to explore:

1. How does the equation $r = 2 \sin \theta$ represent a circle in polar coordinates?
2. What is the general procedure for converting rectangular equations involving trigonometric functions to polar form?
3. How do we interpret $r = 0$ in polar coordinates?
4. What are the differences between polar and rectangular coordinates when graphing conic sections?
5. Can you convert other conic sections (like ellipses or hyperbolas) from rectangular to polar form?

Tip: When converting equations to polar form, always remember the identity $x^2 + y^2 = r^2$, as it simplifies the process!