Which equation represents a circle with a center at $(-3,-5)$ and a radius of 6 units?
A. $(x-3)^2+(y-5)^2=6$
B. $(x-3)^2+(y-5)^2=36$
C. $(x+3)^2+(y+5)^2=6$
D. $(x+3)^2+(y+5)^2=36$
Answer (1)
The problem provides the center and radius of a circle.
We recall the general equation of a circle: ( x − h ) 2 + ( y − k ) 2 = r 2 .
We substitute the given values into the general equation: ( x − ( − 3 ) ) 2 + ( y − ( − 5 ) ) 2 = 6 2 .
We simplify the equation to obtain the final answer: ( x + 3 ) 2 + ( y + 5 ) 2 = 36 .
Explanation
Problem Analysis The problem asks us to find the equation of a circle given its center and radius. We know the general equation of a circle is ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center and r is the radius.
Substitute Values We are given the center ( − 3 , − 5 ) and the radius 6 . We can substitute these values into the general equation of a circle. So, h = − 3 , k = − 5 , and r = 6 .
Form the Equation Substituting the values, we get ( x − ( − 3 ) ) 2 + ( y − ( − 5 ) ) 2 = 6 2 .
Simplify the Equation Simplifying the equation, we have ( x + 3 ) 2 + ( y + 5 ) 2 = 36 .
Final Answer Therefore, the equation of the circle with center ( − 3 , − 5 ) and radius 6 is ( x + 3 ) 2 + ( y + 5 ) 2 = 36 .
Examples
Understanding the equation of a circle is crucial in various fields. For instance, in architecture, engineers use this equation to design circular structures like domes or arches. Imagine designing a circular garden with a sprinkler at its center; the equation helps determine the sprinkler's range to cover the entire garden efficiently, ensuring every plant receives adequate water. This blend of math and practical application enhances both design and functionality.
