Three planets are in circular orbits around a star. Their polar coordinate positions relative to the star are: Location of planet 1 (74.45°) Location of planet 2 (85.74°) Location of planet 3 (93.53°) If the position of the star is (120;45°), which planet is farthest from the star?

1. Convert the polar coordinates of the star to Cartesian coordinates:
(r, \theta) = (120, 45°) \implies (x, y) = (120 \cos 45°, 120 \sin 45°)
x = 120 \cdot \frac{\sqrt{2}}{2} = 60\sqrt{2}
y = 120 \cdot \frac{\sqrt{2}}{2} = 60\sqrt{2}

2. Compare distances from the star to each planet in polar coordinates, remembering that in polar coordinates with the same radius, the further the angle is from the reference angle, the farther it goes:
- Since planet 1 (74.45°) and planet 2 (85.74°), planet 3 (93.53°) all share the same radius defaulting to the polar coordinates, in this simplest problem given note comparing the relative distances each time the largest angle implies the largest relative distance:

Since
93.53° > 85.74° > 74.45°
Answer:
Planet \; 3 (93.53°)