The problem asks to determine the polar coordinates of the point $(-2x, -2y)$ given that the polar coordinates of the point $(x, y)$ are $(r, \theta)$.

Step-by-Step Solution:

1. Polar coordinates basics: For any point $(x, y)$, the polar coordinates $(r, \theta)$ are defined as: $r = \sqrt{x^2 + y^2} \quad \text{and} \quad \theta = \tan^{-1} \left( \frac{y}{x} \right)$

   Therefore, for the given point $(x, y)$, we have: $r = \sqrt{x^2 + y^2} \quad \text{and} \quad \theta = \theta$ (since we are given that the polar coordinates of $(x, y)$ are $(r, \theta)$).

2. New point $(-2x, -2y)$: Now, for the point $(-2x, -2y)$, let's calculate the new polar coordinates.

   • Radius $r'$: The radius is scaled by a factor of 2. Since the new point is a scaled version of the original point by a factor of 2, the new radius $r'$ becomes: $r' = 2r$

   • Angle $\theta'$: The angle will be modified depending on the new point's quadrant. The point $(-2x, -2y)$ is in the opposite quadrant from $(x, y)$, which is a rotation of 180 degrees (or $\pi$ radians) from the original angle $\theta$. Therefore, the new angle $\theta'$ is: $\theta' = \theta + \pi$

Final Polar Coordinates:

Thus, the polar coordinates of the point $(-2x, -2y)$ are: $(2r, \theta + \pi)$

Would you like a detailed explanation or more examples? Let me know if you have any questions!

Here are some related questions to explore further:

1. How do polar coordinates differ from Cartesian coordinates?
2. How do we convert from polar coordinates back to Cartesian coordinates?
3. What happens to the polar angle $\theta$ when the point is reflected across an axis?
4. Can you describe how scaling affects the polar coordinates of a point?
5. How would the polar coordinates change for a point that is translated, rather than scaled?

Tip: When dealing with polar coordinates, always keep track of the quadrant to correctly adjust the angle $\theta$.