Select the correct answer.
Which equation represents a circle with center $T(5,-1)$ and a radius of 16 units?
A. $(x-5)^2+(y+1)^2=16$
B. $(x-5)^2+(y+1)^2=256$
C. $(x+5)^2+(y-1)^2=16$
D. $(x+5)^2+(y-1)^2=256
Answer (1)
The problem provides the center and radius of a circle and asks for its equation.
Recall the standard equation of a circle: ( x − h ) 2 + ( y − k ) 2 = r 2 .
Substitute the given center ( h , k ) = ( 5 , − 1 ) and radius r = 16 into the equation.
Simplify the equation to get the final answer: ( x − 5 ) 2 + ( y + 1 ) 2 = 256 .
Explanation
Problem Analysis We are given a circle with center T ( 5 , − 1 ) and a radius of 16 units. We need to find the equation that represents this circle. The standard equation of a circle with center ( h , k ) and radius r is given by:
Standard Equation of a Circle ( x − h ) 2 + ( y − k ) 2 = r 2
Substitute Values In our case, the center is ( 5 , − 1 ) , so h = 5 and k = − 1 . The radius is r = 16 . Substituting these values into the standard equation, we get:
Equation with Substituted Values ( x − 5 ) 2 + ( y − ( − 1 ) ) 2 = 1 6 2
Simplify the Equation Simplifying the equation, we have:
Simplified Equation ( x − 5 ) 2 + ( y + 1 ) 2 = 256
Compare with Options Comparing this equation with the given options, we see that it matches option B.
Final Answer Therefore, the correct answer is B.
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 helps determine the layout and boundaries. Similarly, in GPS navigation, the equation of a circle can be used to define a search area around a specific location, helping to identify nearby points of interest within a certain radius. This concept is also fundamental in fields like architecture and engineering, where circular shapes are frequently used in designs and constructions.
