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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 \(\theta\).

If the polar coordinates of the point (x, y) are (r, θ), determine the polar coordinates for the point (-2x, -2y).

Explore how to find the polar coordinates of the point (-2x, -2y) given the polar coordinates (r, θ) of the point (x, y). This solution covers the transformation of polar coordinates, focusing on how scaling and quadrant shifts affect radius and angle. Suitable for Grades 10-12.

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