Which of the following could not be points on the unit circle?
A. $\left(\frac{\sqrt{3}}{2}, \frac{1}{3}\right)$
B. $\left(-\frac{2}{3}, \frac{\sqrt{5}}{3}\right)$
C. $(0.8,-0.6)$
D. $(1,1)$
Answer (1)
The equation of the unit circle is x 2 + y 2 = 1 . A point lies on the unit circle if it satisfies this equation.
Point A: ( 2 3 , 3 1 ) gives x 2 + y 2 = 36 31 = 1 .
Point B: ( − 3 2 , 3 5 ) gives x 2 + y 2 = 1 .
Point C: ( 0.8 , − 0.6 ) gives x 2 + y 2 = 1 .
Point D: ( 1 , 1 ) gives x 2 + y 2 = 2 = 1 .
The first point that is not on the unit circle is A.
A
Explanation
Understanding the Unit Circle The equation of the unit circle is x 2 + y 2 = 1 . A point ( x , y ) lies on the unit circle if and only if it satisfies this equation. We are given four points and need to determine which one does not lie on the unit circle.
Checking Each Point Let's check each point:
Point A: ( 2 3 , 3 1 ) . We calculate x 2 + y 2 = ( 2 3 ) 2 + ( 3 1 ) 2 = 4 3 + 9 1 = 36 27 + 36 4 = 36 31 . Since 36 31 = 1 , point A is not on the unit circle.
Point B: ( − 3 2 , 3 5 ) . We calculate x 2 + y 2 = ( − 3 2 ) 2 + ( 3 5 ) 2 = 9 4 + 9 5 = 9 9 = 1 . Since 1 = 1 , point B is on the unit circle.
Point C: ( 0.8 , − 0.6 ) . We calculate x 2 + y 2 = ( 0.8 ) 2 + ( − 0.6 ) 2 = 0.64 + 0.36 = 1 . Since 1 = 1 , point C is on the unit circle.
Point D: ( 1 , 1 ) . We calculate x 2 + y 2 = ( 1 ) 2 + ( 1 ) 2 = 1 + 1 = 2 . Since 2 = 1 , point D is not on the unit circle.
Identifying the Correct Answer Since the question asks for which of the following could NOT be points on the unit circle, and only one answer is expected, we choose the first one that is not on the unit circle, which is point A.
Examples
The unit circle is a fundamental concept in trigonometry and is used extensively in physics, engineering, and computer graphics. For example, when analyzing the motion of a pendulum or the trajectory of a projectile, the unit circle helps to describe the angular displacement and velocity of the object. In computer graphics, the unit circle is used to generate circular shapes and to perform rotations and transformations of objects in 2D space. Understanding the unit circle is crucial for solving problems related to oscillations, waves, and circular motion.
