To find the polar equation of \( xy = 10 \), we use the following relationships between Cartesian and polar coordinates:

\[
x = r\cos\theta \quad \text{and} \quad y = r\sin\theta
\]

Substitute these into the equation \( xy = 10 \):

\[
(r\cos\theta)(r\sin\theta) = 10
\]

Simplify:

\[
r^2\cos\theta\sin\theta = 10
\]

Using the trigonometric identity \( \sin(2\theta) = 2\sin\theta\cos\theta \), rewrite the equation:

\[
r^2 \cdot \frac{\sin(2\theta)}{2} = 10
\]

Multiply through by 2:

\[
r^2\sin(2\theta) = 20
\]

Thus, the polar equation is:

\[
r^2 = \frac{20}{\sin(2\theta)}
\]

This is the polar form of the equation \( xy = 10 \).

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Do you have any questions or would like a deeper explanation?

### Related Questions:
1. How do you convert other Cartesian equations to polar form?
2. What is the physical interpretation of polar equations in coordinate geometry?
3. How does the polar equation \( r^2 = \frac{20}{\sin(2\theta)} \) behave for different values of \( \theta \)?
4. Can polar equations have multiple solutions for a single \( r, \theta \) pair?
5. How does the trigonometric identity \( \sin(2\theta) = 2\sin\theta\cos\theta \) apply to other conversions?

### Tip:
Always remember to check for undefined points in polar equations, such as where the denominator becomes zero.

Discover how to convert the Cartesian equation xy=10 into its polar form step by step. This problem explores key concepts in polar coordinates and trigonometry, using relationships between Cartesian and polar systems. Suitable for high school students in Grades 10-12.

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