Which equation represents a circle with a center at $(2,-8)$ and a radius of 11?
$(x-8)^2+(y+2)^2=11$
$(x-2)^2+(y+8)^2=121$
$(x+2)^2+(y-8)^2=11$
$(x+8)^2+(y-2)^2=121
Answer (1)
The problem provides the center and radius of a circle.
The general equation of a circle is ( x − a ) 2 + ( y − b ) 2 = r 2 , where ( a , b ) is the center and r is the radius.
Substitute the given center ( 2 , − 8 ) and radius 11 into the general equation.
Simplify the equation to get the final answer: ( 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 form of a circle's equation and can substitute the given values to find the correct equation.
General Equation of a Circle The general equation of a circle with center ( a , b ) and radius r is given by: ( x − a ) 2 + ( y − b ) 2 = r 2
Substitute Given Values We are given the center ( 2 , − 8 ) and radius 11 . Substituting these values into the general equation, we get: ( x − 2 ) 2 + ( y − ( − 8 ) ) 2 = 1 1 2
Simplify the Equation Simplifying the equation, we have: ( x − 2 ) 2 + ( y + 8 ) 2 = 121
Identify the Correct Option 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 useful in many real-world applications. For example, civil engineers use it when designing circular structures like tunnels or roundabouts. Imagine you're designing a circular roundabout with a specific center point and radius to optimize traffic flow. The equation of a circle helps you define the exact boundaries of the roundabout, ensuring it fits perfectly within the available space and meets all design requirements. Also, in computer graphics, circles are fundamental shapes used in creating various visual elements, from simple icons to complex game environments.
