Questions: convert the polar equation to rectangular form and sketch the graph. r = 6 cos θ

Solution Steps

The given polar equation is: \[ r = 6 \cos \theta \]

Using the relationships \( x = r \cos \theta \) and \( r = \sqrt{x^2 + y^2} \), we can convert this to rectangular form.

First, multiply both sides by \( r \): \[ r^2 = 6r \cos \theta \]

Since \( r \cos \theta = x \): \[ r^2 = 6x \]

And since \( r^2 = x^2 + y^2 \): \[ x^2 + y^2 = 6x \]

Rearrange to get the standard form of a circle: \[ x^2 + y^2 - 6x = 0 \]

Complete the square for the \( x \) terms: \[ (x^2 - 6x + 9) + y^2 = 9 \] \[ (x - 3)^2 + y^2 = 3^2 \]

This is the equation of a circle with center at \( (3, 0) \) and radius 3.

Final Answer