Convert the following rectangular equation to polar form, showing the work. x^2+y^2=16x

To convert the rectangular equation \( x^2 + y^2 = 16x \) to polar form, we start by using the following polar-coordinate relationships:

1. \( x = r \cos \theta \)
2. \( y = r \sin \theta \)
3. \( x^2 + y^2 = r^2 \)

Let's go through the steps:

### Step 1: Substitute \( x^2 + y^2 = r^2 \)

The equation \( x^2 + y^2 = 16x \) becomes:
\[
r^2 = 16x
\]

### Step 2: Substitute \( x = r \cos \theta \)

Now, substitute \( x = r \cos \theta \) into the equation:
\[
r^2 = 16(r \cos \theta)
\]

### Step 3: Simplify the equation

Now, divide both sides by \( r \) (assuming \( r \neq 0 \)):
\[
r = 16 \cos \theta
\]

### Final Polar Form

The polar form of the given equation is:
\[
r = 16 \cos \theta
\]

This is the equivalent equation in polar coordinates.

Would you like further details or have any questions about the steps?

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Here are five related questions to deepen your understanding:

1. What would the polar form be if the equation were \( x^2 + y^2 = 16y \)?
2. How would you convert \( x^2 - y^2 = 16x \) to polar form?
3. What is the general process for converting a rectangular equation to polar form?
4. How would the graph of \( r = 16 \cos \theta \) look in polar coordinates?
5. Can this polar form equation be converted back to rectangular form?

**Tip:** Remember that \( x^2 + y^2 = r^2 \) is a useful identity when converting rectangular equations involving \( x \) and \( y \) to polar coordinates.

Convert the following rectangular equation to polar form, showing the work. x^2 + y^2 = 16x

Learn how to convert the rectangular equation x^2 + y^2 = 16x to polar form. This guide provides a step-by-step approach, explaining key transformations and formulas used in polar coordinates. Ideal for high school students in Grades 10-12.

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