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 $r$.