Which equation represents a circle with a center at $(2,-8)$ and a radius of 11?
A. $(x-8)^2+(y+2)^2=11$
B. $(x-2)^2+(y+8)^2=121$
C. $(x+2)^2+(y-8)^2=11$
D. $(x+8)^2+(y-2)^2=121
Answer (1)
The equation of a circle with center ( h , k ) and radius r is ( x − h ) 2 + ( y − k ) 2 = r 2 . Substituting the given center ( 2 , − 8 ) and radius 11 into the equation, we get ( x − 2 ) 2 + ( y − ( − 8 ) ) 2 = 1 1 2 . Simplifying, we find the equation to be ( x − 2 ) 2 + ( y + 8 ) 2 = 121 . Therefore, the equation representing the circle is ( x − 2 ) 2 + ( y + 8 ) 2 = 121 .
Explanation
Problem Analysis The problem asks us to identify the equation of a circle given its center and radius. We know the general equation of a circle with center ( h , k ) and radius r is ( x − h ) 2 + ( y − k ) 2 = r 2 . We are given the center ( 2 , − 8 ) and the radius 11 .
Substitution We need to substitute the given values into the general equation of a circle. The center is ( h , k ) = ( 2 , − 8 ) and the radius is r = 11 . Thus, the equation becomes ( x − 2 ) 2 + ( y − ( − 8 ) ) 2 = 1 1 2 .
Simplification Now, we simplify the equation. We have ( x − 2 ) 2 + ( y + 8 ) 2 = 121 .
Final Answer Comparing this equation with the given options, we find that the correct equation is ( x − 2 ) 2 + ( y + 8 ) 2 = 121 .
Examples
Understanding the equation of a circle is crucial in various real-world applications. For instance, when designing a circular garden, knowing the center and radius helps determine the layout and boundaries. Similarly, in GPS navigation, the equation of a circle can be used to define a specific range or area around a location. This concept is also fundamental in fields like architecture and engineering, where circular shapes are frequently used in designs and constructions.
