Rewrite the curve $$ x^2+y^2=4x\text{ in polar coordinates} $$

To rewrite the curve $ x^2 + y^2 = 4x $ in polar coordinates, we can use the relationships $ x = r \cos(\theta) $ and $ y = r \sin(\theta) $. 

Starting with the given equation:

\[
x^2 + y^2 = 4x
\]

Substituting the polar coordinates:

\[
(r \cos(\theta))^2 + (r \sin(\theta))^2 = 4(r \cos(\theta))
\]

This simplifies to:

\[
r^2 (\cos^2(\theta) + \sin^2(\theta)) = 4r \cos(\theta)
\]

Using the Pythagorean identity $ \cos^2(\theta) + \sin^2(\theta) = 1 $, we have:

\[
r^2 = 4r \cos(\theta)
\]

Dividing both sides by $ r $ (assuming $ r \neq 0 $) gives:

\[
r = 4 \cos(\theta)
\]

Thus, the curve in polar coordinates is:

\[
r = 4 \cos(\theta)
\]

Would you like to explore more about this curve or have any questions? Here are some related questions you might find interesting:

1. What does the graph of $ r = 4 \cos(\theta) $ look like?
2. How do you derive polar coordinates from Cartesian coordinates generally?
3. What is the significance of the curve being a circle?
4. How does this curve behave as $ \theta $ varies?
5. Can you convert other conic sections to polar coordinates? 

Tip: Always check the domain of $ \theta $ when working with polar equations!

Rewrite the curve $$ x^2+y^2=4x $$ in polar coordinates.

Discover how to convert the curve x^2 + y^2 = 4x into polar coordinates with a step-by-step solution. This topic covers important concepts in polar coordinates and analytic geometry. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation