Question

The polar co-ordinates of the point whose cartesian co-ordinates are \( (-2, -2) \), are given by

• A. \( \left( 2\sqrt{2}, \frac{5\pi}{4} \right) \)
• B. \( \left( 2\sqrt{2}, \frac{3\pi}{4} \right) \)
• C. \( \left( 2\sqrt{2}, \frac{7\pi}{6} \right) \)
• D. \( \left( 2\sqrt{2}, \frac{\pi}{4} \right) \)

Hint:

To convert from cartesian to polar coordinates, use the formulas \( r = \sqrt{x^2 + y^2} \) and \( \theta = \tan^{-1} \left( \frac{y}{x} \right) \).

Answer 1

The Correct Option is A

Step 1: Converting to polar coordinates.
The polar coordinates \( (r, \theta) \) are related to the cartesian coordinates \( (x, y) \) by: \[ r = \sqrt{x^2 + y^2}, \quad \theta = \tan^{-1} \left( \frac{y}{x} \right) \] For the given point \( (-2, -2) \), we calculate: \[ r = \sqrt{(-2)^2 + (-2)^2} = \sqrt{8} = 2\sqrt{2} \] \[ \theta = \tan^{-1} \left( \frac{-2}{-2} \right) = \frac{5\pi}{4} \]
Step 2: Conclusion.
Thus, the polar coordinates of the point are \( \left( 2\sqrt{2}, \frac{5\pi}{4} \right) \), which makes option (A) the correct answer.