To solve the problem of converting the rectangular coordinates $(1, -1)$ into polar coordinates $(r, \theta)$ with the conditions $r \leq 0$ and $0 \leq \theta \leq 2\pi$, we follow these steps:

Step 1: Compute $r$

The formula for the radial coordinate $r$ is: $r = \pm \sqrt{x^2 + y^2}$ Here, $x = 1$ and $y = -1$, so: $r = \pm \sqrt{1^2 + (-1)^2} = \pm \sqrt{1 + 1} = \pm \sqrt{2}$ Since the condition is $r \leq 0$, we take $r = -\sqrt{2}$.

Step 2: Compute $\theta$

The angle $\theta$ is calculated using: $\tan(\theta) = \frac{y}{x}$ Here, $\frac{y}{x} = \frac{-1}{1} = -1$.

The reference angle where $\tan(\theta) = 1$ is $\frac{\pi}{4}$, but since the point $(1, -1)$ is in the fourth quadrant, the angle is: $\theta = 2\pi - \frac{\pi}{4} = \frac{7\pi}{4}$

Step 3: Final Polar Coordinates

The polar coordinates are: $(r, \theta) = (-\sqrt{2}, \frac{7\pi}{4})$

Step 4: Verify the Conditions

• $r \leq 0$: True, as $r = -\sqrt{2}$.
• $0 \leq \theta \leq 2\pi$: True, as $\theta = \frac{7\pi}{4}$.

The correct answer is: $\text{a) } \left(-\sqrt{2}, \frac{7\pi}{4}\right)$

Let me know if you'd like further clarification or additional questions. Here are some related questions:

1. What is the process for converting polar coordinates back to rectangular form?
2. How do we determine the quadrant of a point in polar coordinates?
3. How does the sign of $r$ affect the representation of polar coordinates?
4. Can two different polar coordinates represent the same point? Explain why.
5. Why is $\tan^{-1}$ used to find $\theta$, and how do we determine the correct quadrant?

Tip: When working with polar coordinates, always carefully consider the signs of $r$ and $\theta$ and their impact on the location of the point.