Select the correct answer.
Which equation represents a circle with a center at $(-5,5)$ and a radius of 3 units?
A. $(x+5)^2+(y-5)^2=3$
B. $(x+5)^2+(y-5)^2=9$
C. $(x+5)^2+(y-5)^2=6$
D. $(x-5)^2+(y+5)^2=9$
E. $(x-5)^2+(y+5)^2=3
Answer (1)
Start with the general equation of a circle: ( x − h ) 2 + ( y − k ) 2 = r 2 .
Substitute the given center ( − 5 , 5 ) for ( h , k ) and the radius 3 for r .
Simplify the equation: ( x + 5 ) 2 + ( y − 5 ) 2 = 9 .
The equation of the circle is ( x + 5 ) 2 + ( y − 5 ) 2 = 9 .
Explanation
Analyze the problem and given data The problem asks us to identify 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 ( x − h ) 2 + ( y − k ) 2 = r 2 . We are given the center ( − 5 , 5 ) and the radius 3 .
Substitute the given values Now, we substitute the given values into the general equation of a circle:
( x − ( − 5 ) ) 2 + ( y − 5 ) 2 = 3 2
Simplify the equation Simplify the equation:
( x + 5 ) 2 + ( y − 5 ) 2 = 9
Select the correct answer Comparing this equation with the given options, we find that option B matches our result.
Therefore, the correct answer is B.
Examples
Understanding the equation of a circle is very useful in various 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 central statue. If you place the statue at coordinates (-5, 5) on a map and want the roundabout to have a radius of 3 units, the equation (x+5)^2 + (y-5)^2 = 9 helps define the exact boundaries of the roundabout, ensuring it fits perfectly within the available space and that all points on the edge of the roundabout are exactly 3 units away from the statue.
