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The image involves converting polar coordinates with a positive radius to a form with a negative radius. Specifically, the transformation being discussed is based on the principle that for a negative radius \( r \), the corresponding angle \( \theta \) must be shifted by \( \pi \) radians, since \( (-r, \theta) \) is equivalent to \( (r, \theta + \pi) \).

Given that the original polar coordinates are \( (r, \theta) = \left(7\sqrt{2}, \frac{7\pi}{4}\right) \), and you want the corresponding coordinates with a negative radius \( r = -7\sqrt{2} \), the new angle will be:

\[
\theta' = \frac{7\pi}{4} + \pi = \frac{7\pi}{4} + \frac{4\pi}{4} = \frac{11\pi}{4}
\]

Since \( \frac{11\pi}{4} \) is greater than \( 2\pi \), we subtract \( 2\pi \) to keep the angle in the range \( [0, 2\pi) \):

\[
\frac{11\pi}{4} - 2\pi = \frac{11\pi}{4} - \frac{8\pi}{4} = \frac{3\pi}{4}
\]

Thus, the polar coordinates with the negative radius are:

\[
(r, \theta) = \left(-7\sqrt{2}, \frac{3\pi}{4}\right)
\]

Would you like further clarification on this? Here are 5 related questions for practice:

1. How do you convert polar coordinates with positive \( r \) to negative \( r \)?
2. What is the relationship between \( r \) and \( \theta \) when switching the radius sign?
3. How do you express an angle greater than \( 2\pi \) in the range \( [0, 2\pi) \)?
4. What are the polar coordinates of a point if the original angle is \( \frac{5\pi}{6} \) and the radius becomes negative?
5. How do you interpret polar coordinates geometrically when the radius is negative?

**Tip**: Always ensure the angle \( \theta \) in polar coordinates is within the standard interval \( [0, 2\pi) \) when switching the sign of \( r \).

Given the polar coordinates (r, θ) = (7√2, 7π/4), find the polar coordinates with a negative radius r = -7√2.

Learn how to convert polar coordinates (r, θ) = (7√2, 7π/4) to a negative radius. This step-by-step solution covers the concept of polar coordinates transformation and angle adjustment. Suitable for students in Grade 11-12.

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