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

The equation of a circle with center ( h , k ) and radius r is ( x − h ) 2 + ( y − k ) 2 = r 2 . Given the center ( 2 , − 8 ) and radius 11 , we substitute these values into the equation. We have:

Substitute the center and radius: ( x − 2 ) 2 + ( y − ( − 8 ) ) 2 = 1 1 2 .
Simplify the equation: ( x − 2 ) 2 + ( y + 8 ) 2 = 121 .
The equation of the circle is ( x − 2 ) 2 + ( y + 8 ) 2 = 121 ​ .

Explanation

Problem Analysis The problem asks us to find the equation of a circle given its center and radius. We know that the general equation of a circle with center ( h , k ) and radius r is given by:

General Equation of a Circle ( x − h ) 2 + ( y − k ) 2 = r 2

Substitute Values In this problem, we are given the center ( h , k ) = ( 2 , − 8 ) and the radius r = 11 . We can substitute these values into the general equation of a circle:

Equation with Substituted Values ( x − 2 ) 2 + ( y − ( − 8 ) ) 2 = 1 1 2

Simplify the Equation Simplifying the equation, we get:

Simplified Equation ( x − 2 ) 2 + ( y + 8 ) 2 = 121

Final Answer Therefore, the equation of the circle with center ( 2 , − 8 ) and radius 11 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 allows you to determine the exact placement and size of the garden using the equation of a circle. Similarly, in architecture, circular arches and domes can be precisely planned using this equation, ensuring structural integrity and aesthetic appeal. The equation also finds use in computer graphics for drawing circles and circular arcs.

Answered by GinnyAnswer | 2025-07-08