Which point is on the circle centered at the origin with a radius of 5 units?
Distance formula: $\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$
A. $(2, \sqrt{21})$
B. $(2, \sqrt{23})$
C. $(2,1)$
D. $(2,3)$
Answer (1)
Analyze the problem: Determine which point lies on a circle centered at the origin with a radius of 5.
Apply the distance formula: Calculate the distance of each point from the origin using x 2 + y 2 .
Check the distance: Verify if the calculated distance equals the radius (5).
Identify the point on the circle: Conclude that the point ( 2 , 21 ) lies on the circle because its distance from the origin is 5. ( 2 , 21 )
Explanation
Problem Analysis We are given a circle centered at the origin (0, 0) with a radius of 5. We need to determine which of the given points lies on this circle. A point lies on the circle if its distance from the origin is equal to the radius. We will use the distance formula to calculate the distance of each point from the origin and check if it equals 5.
Distance Formula The distance formula is given by ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 . Since the circle is centered at the origin, ( x 1 , y 1 ) = ( 0 , 0 ) . Thus, the distance formula simplifies to x 2 + y 2 , where ( x , y ) are the coordinates of the point we are testing.
Testing Point (2, sqrt(21)) Let's test the point ( 2 , 21 ) . The distance from the origin is 2 2 + ( 21 ) 2 = 4 + 21 = 25 = 5 . Since the distance is 5, this point lies on the circle.
Testing Point (2, sqrt(23)) Let's test the point ( 2 , 23 ) . The distance from the origin is 2 2 + ( 23 ) 2 = 4 + 23 = 27 ≈ 5.196 . Since the distance is not 5, this point does not lie on the circle.
Testing Point (2, 1) Let's test the point ( 2 , 1 ) . The distance from the origin is 2 2 + 1 2 = 4 + 1 = 5 ≈ 2.236 . Since the distance is not 5, this point does not lie on the circle.
Testing Point (2, 3) Let's test the point ( 2 , 3 ) . The distance from the origin is 2 2 + 3 2 = 4 + 9 = 13 ≈ 3.606 . Since the distance is not 5, this point does not lie on the circle.
Conclusion Therefore, the point ( 2 , 21 ) is the only point that lies on the circle centered at the origin with a radius of 5.
Examples
Circles are fundamental in many real-world applications, from designing gears and wheels to understanding satellite orbits. For instance, if you're designing a Ferris wheel with a specific radius, you need to ensure that the cabins are placed precisely on the circumference to guarantee a smooth and safe ride. Similarly, in GPS technology, satellites orbit the Earth in circular paths, and knowing their exact position on the orbit (a circle) is crucial for accurate location tracking.
