A circle centered at the origin contains the point $(0,-9)$. Does $(8, \sqrt{17})$ also lie on the circle?
No, the distance from the center to the point ($8, \sqrt{17}$) is not the same as the radius.
No, the radius of 10 units is different from the distance from the center to the point $(8, \sqrt{17})$.
Yes, the distance from the origin to the point ($8, \sqrt{17}$) is 9 units.
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 ) , which is 9.
Calculate the distance from the origin to the point ( 8 , 17 ) , which is also 9.
Compare the distance to the radius.
Conclude that since the distance equals the radius, the point ( 8 , 17 ) lies on the circle: Yes .
Explanation
Problem Analysis The problem states that a circle is centered at the origin and contains the point ( 0 , − 9 ) . We need to determine if the point ( 8 , 17 ) also lies on the circle. To do this, we need to find the radius of the circle and then check if the distance from the origin to the point ( 8 , 17 ) is equal to the radius.
Calculate the Radius First, we find the radius of the circle. Since the circle is centered at the origin ( 0 , 0 ) and contains the point ( 0 , − 9 ) , the radius is the distance between these two points. We use the distance formula: r = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 In this case, ( x 1 , y 1 ) = ( 0 , 0 ) and ( x 2 , y 2 ) = ( 0 , − 9 ) . So, r = ( 0 − 0 ) 2 + ( − 9 − 0 ) 2 = 0 2 + ( − 9 ) 2 = 81 = 9 Thus, the radius of the circle is 9 units.
Calculate the Distance Next, we calculate the distance from the origin ( 0 , 0 ) to the point ( 8 , 17 ) . Again, we use the distance formula: d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 In this case, ( x 1 , y 1 ) = ( 0 , 0 ) and ( x 2 , y 2 ) = ( 8 , 17 ) . So, d = ( 8 − 0 ) 2 + ( 17 − 0 ) 2 = 8 2 + ( 17 ) 2 = 64 + 17 = 81 = 9 Thus, the distance from the origin to the point ( 8 , 17 ) is 9 units.
Conclusion Since the distance from the origin to the point ( 8 , 17 ) is equal to the radius of the circle (both are 9 units), the point ( 8 , 17 ) lies on the circle.
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 circular park and want to place a statue a certain distance from the center, knowing the equation of a circle helps ensure the statue lies within the park's boundary. Similarly, in astronomy, understanding circular orbits helps predict the positions of celestial bodies.
