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 (2)
The problem provides the center ( − 3 , − 5 ) and radius 6 of a circle.
The student recalls the standard equation of a circle: ( x − h ) 2 + ( y − k ) 2 = r 2 .
The student substitutes the given values into the equation: ( x − ( − 3 ) ) 2 + ( y − ( − 5 ) ) 2 = 6 2 .
The student simplifies the equation to get the final answer: ( x + 3 ) 2 + ( y + 5 ) 2 = 36 .
Explanation
Problem Analysis The problem asks us to identify the equation of a circle given its center and radius. We know the center is at ( − 3 , − 5 ) and the radius is 6 units. The standard form of a circle's equation is ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center and r is the radius.
Substitution and Simplification Now, we substitute the given values into the standard equation. The center ( h , k ) is ( − 3 , − 5 ) , so h = − 3 and k = − 5 . The radius r is 6. Plugging these values into the equation, we get:
( x − ( − 3 ) ) 2 + ( y − ( − 5 ) ) 2 = 6 2
Simplifying this, we have:
( x + 3 ) 2 + ( y + 5 ) 2 = 36
Identifying the Correct Equation Comparing our derived equation ( x + 3 ) 2 + ( y + 5 ) 2 = 36 with the given options, we find that it matches the fourth option.
Final Answer Therefore, the equation that represents a circle with a center at ( − 3 , − 5 ) and a radius of 6 units is ( x + 3 ) 2 + ( y + 5 ) 2 = 36 .
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 windows or domes can be precisely planned using this equation, ensuring structural integrity and aesthetic appeal. In navigation, the equation of a circle can help define the range of a radar system or the area covered by a cell tower, providing essential information for coverage planning and signal analysis.
The equation of a circle with center at (-3, -5) and radius of 6 is (x + 3)^2 + (y + 5)^2 = 36. This matches option D from the given choices. Therefore, the answer is D.
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