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 .
Substitute the given center ( 2 , − 8 ) and radius 11 into the equation.
Simplify the equation to get ( x − 2 ) 2 + ( y + 8 ) 2 = 121 .
The equation representing the circle is ( x − 2 ) 2 + ( y + 8 ) 2 = 121 .
Explanation
Problem Analysis The equation of a circle with center ( h , k ) and radius r is given by ( x − h ) 2 + ( y − k ) 2 = r 2 . In this problem, we are given the center ( 2 , − 8 ) and the radius 11 . We need to find the equation of the circle.
Substitution We substitute the given values h = 2 , k = − 8 , and r = 11 into the equation of a circle: ( x − 2 ) 2 + ( y − ( − 8 ) ) 2 = 1 1 2 ( x − 2 ) 2 + ( y + 8 ) 2 = 121
Comparison Comparing the derived 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, consider designing a circular garden with a specific center and radius. The equation of the circle helps define the boundary of the garden, ensuring accurate placement and dimensions. Similarly, in architecture, circular windows or domes can be precisely designed using the circle equation, guaranteeing structural integrity and aesthetic appeal. The equation also finds use in computer graphics for drawing circles and arcs, essential for creating various visual elements.
