Determine two pairs of polar coordinates for the point with 0° ≤θ

Question

Determine two pairs of polar coordinates for the point with 0° ≤θ < 360°.
-$( 0 , - \sqrt { 5 } )$
A) $\left( \sqrt { 5 } , 270 ^ { \circ } \right) , \left( - \sqrt { 5 } , 90 ^ { \circ } \right)$
B) $\left( \sqrt { 5 } , 0 ^ { \circ } \right) , \left( - \sqrt { 5 } , 180 ^ { \circ } \right)$
C) $\left( \sqrt { 5 } , 90 ^ { \circ } \right) , \left( - \sqrt { 5 } , 0 ^ { \circ } \right)$
D) $\left( \sqrt { 5 } , 90 ^ { \circ } \right) , \left( - \sqrt { 5 } , 270 ^ { \circ } \right)$

Quizplus · Accepted Answer

The point $(0, -\sqrt{5})$ is located on the negative y-axis. The polar coordinates are determined by the distance from the origin and the angle from the positive x-axis. The distance from the origin is $\sqrt{5}$, and the angle to the negative y-axis is $270^\circ$. Thus, $\left(\sqrt{5}, 270^\circ\right)$ is correct. The alternative representation using a negative radius flips the angle by $180^\circ$, making it $90^\circ$, but this option is not correctly represented in any of the choices, indicating a misunderstanding in the formulation of the question or choices. The correct interpretation of polar coordinates for the given point is indeed $\left(\sqrt{5}, 270^\circ\right)$, but the second pair in option A is not conventionally correct in standard polar coordinate representation, as negative radii are less common and typically the angle is adjusted to keep the radius positive.

Determine Two Pairs of Polar Coordinates for the Point with 0°