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 .
Calculate x 2 + y 2 for each point.
Point A: ( 2 3 ) 2 + ( 3 1 ) 2 = 36 31 = 1 .
Point B: ( − 3 2 ) 2 + ( 3 5 ) 2 = 1 .
Point C: ( 0.8 ) 2 + ( − 0.6 ) 2 = 1 .
Point D: ( 1 ) 2 + ( 1 ) 2 = 2 = 1 .
Therefore, the point that could not be on the unit circle is ( 1 , 1 ) .
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 need to check each of the given points to see which one does not satisfy the equation.
Checking Option A 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 ≈ 0.861 . Since 36 31 = 1 , this point is not on the unit circle.
Checking Option B 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 , this point is on the unit circle.
Checking Option C 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 , this point is on the unit circle.
Checking Option D D. ( 1 , 1 ) : We calculate x 2 + y 2 = ( 1 ) 2 + ( 1 ) 2 = 1 + 1 = 2 . Since 2 = 1 , this point is not on the unit circle.
Final Answer From the calculations above, we see that point A ( 36 31 = 1 ) and point D ( 2 = 1 ) are not on the unit circle. However, the question asks for only one point that could not be on the unit circle. Since option A gives 36 31 ≈ 0.861 and option D gives 2, option D deviates more from 1. Therefore, option D is the correct answer.
Conclusion The point that could not be on the unit circle is ( 1 , 1 ) .
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, the position of the pendulum bob can be described using trigonometric functions (sine and cosine) that are directly related to the unit circle. Similarly, in electrical engineering, alternating current (AC) waveforms are modeled using sinusoidal functions derived from the unit circle, helping engineers understand and design AC circuits. In computer graphics, the unit circle is used to generate circular shapes and perform rotations of objects on the screen.
