A circle centered at the origin contains the point $(0,-9)$. Does $(8, \sqrt{17})$ also lie on the circle?
A. No, the distance from the center to the point ($8, \sqrt{17}$) is not the same as the radius.
B. No, the radius of 9 units is different from the distance from the center to the point $(8, \sqrt{17})$.
C. Yes, the distance from the origin to the point ($8, \sqrt{17}$) is 9 units.
D. Yes, the distance from the point $(0,-9)$ to the point ($8, \sqrt{17}$) is 9 units.
Answer (1)
Calculate the radius of the circle using the point (0, -9): r = ( 0 − 0 ) 2 + ( − 9 − 0 ) 2 = 9 .
Calculate the distance between the origin and the point (8, \sqrt{17}): d = ( 8 − 0 ) 2 + ( 17 − 0 ) 2 = 64 + 17 = 9 .
Compare the distance with the radius: Since d = r = 9 , the point (8, \sqrt{17}) lies on the circle.
Conclude that the point (8, \sqrt{17}) lies on the circle: Yes, the distance from the origin to the point ( 8 , 17 ) is 9 units.
Explanation
Problem Analysis The problem states that a circle is centered at the origin (0,0) and contains the point (0, -9). We need to determine if the point (8, \sqrt{17}) also lies on the circle. To do this, we will first find the radius of the circle and then calculate the distance between the origin and the point (8, \sqrt{17}). If this distance is equal to the radius, then the point lies on the circle.
Calculate the Radius First, let's find the radius of the circle. Since the circle is centered at the origin and contains the point (0, -9), the radius is the distance between these two points. Using the distance formula: r = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 r = ( 0 − 0 ) 2 + ( − 9 − 0 ) 2 r = 0 2 + ( − 9 ) 2 r = 81 r = 9 So, the radius of the circle is 9 units.
Calculate the Distance Next, we need to calculate the distance between the origin (0,0) and the point (8, \sqrt{17}). Again, using the distance formula: d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 d = ( 8 − 0 ) 2 + ( 17 − 0 ) 2 d = 8 2 + ( 17 ) 2 d = 64 + 17 d = 81 d = 9 The distance between the origin and the point (8, \sqrt{17}) is 9 units.
Conclusion Since the distance between the origin and the point (8, \sqrt{17}) is 9 units, which is equal to the radius of the circle, the point (8, \sqrt{17}) lies on the circle.
Therefore, the correct answer is: Yes, the distance from the origin to the point (8, \sqrt{17}) is 9 units.
Examples
Circles are fundamental in many real-world applications, from designing gears and wheels to understanding satellite orbits. For example, if you're designing a circular garden and want to place a sprinkler at the center, knowing the radius helps determine the sprinkler's coverage area. Similarly, in astronomy, understanding the orbits of planets and satellites relies on the properties of circles and ellipses, allowing us to predict their positions and movements.
