Circle [tex]$C$[/tex] is centered at the origin. If [tex]$Q(10,0)$[/tex] lies on circle [tex]$C$[/tex], which of the following points also lies on circle [tex]$C$[/tex]?
A. [tex]$(4,5 \sqrt{3})$[/tex]
B. [tex]$(5,5 \sqrt{3})$[/tex]
C. [tex]$(3, \sqrt{34})$[/tex]
D. [tex]$(6,4)$[/tex]
Answer (1)
The radius of circle C is found using point Q ( 10 , 0 ) , resulting in r = 10 .
The equation of circle C centered at the origin is x 2 + y 2 = 100 .
Each point is tested to see if it satisfies the circle's equation.
Point ( 5 , 5 3 ) satisfies the equation, therefore the answer is ( 5 , 5 3 ) .
Explanation
Analyze the problem The problem states that circle C is centered at the origin ( 0 , 0 ) and that the point Q ( 10 , 0 ) lies on the circle. We need to determine which of the given points also lies on circle C .
Determine the radius Since Q ( 10 , 0 ) lies on the circle centered at the origin, the radius of the circle is the distance between the origin and Q . We can calculate the radius r using the distance formula: r = ( 10 − 0 ) 2 + ( 0 − 0 ) 2 = 100 = 10 Thus, the radius of the circle is 10.
State the circle's equation The equation of a circle centered at the origin is given by x 2 + y 2 = r 2 . Since r = 10 , the equation of circle C is: x 2 + y 2 = 1 0 2 = 100
Check each point Now we need to check each of the given points to see if they satisfy the equation of the circle.
A. ( 4 , 5 3 ) : 4 2 + ( 5 3 ) 2 = 16 + 25 ( 3 ) = 16 + 75 = 91 Since 91 = 100 , point A does not lie on the circle.
B. ( 5 , 5 3 ) : 5 2 + ( 5 3 ) 2 = 25 + 25 ( 3 ) = 25 + 75 = 100 Since 100 = 100 , point B lies on the circle.
C. ( 3 , 34 ) : 3 2 + ( 34 ) 2 = 9 + 34 = 43 Since 43 = 100 , point C does not lie on the circle.
D. ( 6 , 4 ) : 6 2 + 4 2 = 36 + 16 = 52 Since 52 = 100 , point D does not lie on the circle.
Conclusion The only point that satisfies the equation of the circle x 2 + y 2 = 100 is ( 5 , 5 3 ) . Therefore, the point that lies on circle C is ( 5 , 5 3 ) .
Examples
Understanding circles and their equations is crucial in many real-world applications. For example, when designing a circular garden, you need to know the equation of the circle to determine the placement of plants at specific coordinates. Similarly, in architecture, circular arches and domes require precise calculations based on the circle's equation to ensure structural integrity and aesthetic appeal. This knowledge also extends to fields like astronomy, where the orbits of planets and satellites can be approximated as circles or ellipses, and their positions can be determined using similar mathematical principles.
