A circle centered at the origin contains the point $(0,-9)$. Does $(8, \sqrt{17})$ also lie on the circle? Explain.
Answer (2)
Determine the radius of the circle using the given point ( 0 , − 9 ) : r = 9 .
Calculate the distance from the origin to the point ( 8 , s q r t 17 ) : d = 9 .
Compare the distance to the radius.
Since the distance equals the radius, the point ( 8 , s q r t 17 ) lies on the circle: Yes .
Explanation
Problem Analysis The problem asks us to determine if the point ( 8 , s q r t 17 ) lies on a circle centered at the origin that also contains the point ( 0 , − 9 ) . 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 , s q r t 17 ) is equal to the radius.
Finding the Radius First, let's 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 can use the distance formula: r = s q r t ( 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 = s q r t ( 0 − 0 ) 2 + ( − 9 − 0 ) 2 = s q r t 0 + ( − 9 ) 2 = s q r t 81 = 9 Thus, the radius of the circle is 9.
Finding the Distance Next, we need to find the distance from the origin ( 0 , 0 ) to the point ( 8 , s q r t 17 ) . Again, we use the distance formula: d = s q r t ( 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 , s q r t 17 ) . So, d = s q r t ( 8 − 0 ) 2 + ( s q r t 17 − 0 ) 2 = s q r t 8 2 + ( s q r t 17 ) 2 = s q r t 64 + 17 = s q r t 81 = 9 Thus, the distance from the origin to the point ( 8 , s q r t 17 ) is 9.
Conclusion Since the distance from the origin to the point ( 8 , s q r t 17 ) is equal to the radius of the circle (both are 9), the point ( 8 , s q r t 17 ) lies on the circle.
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 you determine the sprinkler's range to cover the entire garden. Similarly, in astronomy, understanding the orbits of planets and satellites relies on the properties of circles and ellipses, allowing scientists to predict their positions accurately.
The radius of the circle is 9, and the distance from the origin to the point ( 8 , 17 ) is also 9. Since these values are equal, the point lies on the circle. Therefore, the answer is yes.
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