The task in the image is to convert polar equations to rectangular form. To do this, we use the following standard relationships between polar and rectangular coordinates:

$x = r \cos \theta, \quad y = r \sin \theta, \quad r^2 = x^2 + y^2$

Let's address each part:

Part (a) $r \cos \theta = 2$

Using the conversion formula $x = r \cos \theta$, we substitute this into the equation:

$x = 2$

This is the rectangular form of the equation.

Part (b) $r = -6$

We use the relationship $r^2 = x^2 + y^2$ to convert this into rectangular form. First, square both sides of the equation:

$r^2 = (-6)^2 = 36$

Now substitute $r^2 = x^2 + y^2$:

$x^2 + y^2 = 36$

This is the rectangular form of the equation.

Let me know if you need further clarification or details.

Here are 5 related questions you might find interesting:

1. What happens to the rectangular form if we change the angle $\theta$ in polar coordinates?
2. How would the equation look if the polar form had $r \sin \theta$?
3. Can we convert rectangular coordinates back to polar coordinates easily?
4. What is the significance of negative $r$ in polar coordinates?
5. How would we plot these equations on a graph?

Tip:

Always remember that $\theta$ in polar coordinates represents the angle with the positive $x$-axis, and adjusting it modifies the direction of the vector from the origin!