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.