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)
Recall the standard equation of a circle: ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center and r is the radius.
Substitute the given center ( 5 , − 1 ) and radius 16 into the equation: ( x − 5 ) 2 + ( y − ( − 1 ) ) 2 = 1 6 2 .
Simplify the equation: ( x − 5 ) 2 + ( y + 1 ) 2 = 256 .
The correct equation is ( x − 5 ) 2 + ( y + 1 ) 2 = 256 .
Explanation
Analyze the problem and recall the circle equation. The problem asks us to identify the equation of a circle given its center and radius. We know the center is at T ( 5 , − 1 ) and the radius is 16 units. The standard equation of a circle with center ( h , k ) and radius r is ( x − h ) 2 + ( y − k ) 2 = r 2 . We need to substitute the given values into this equation and simplify.
Substitute the values into the equation. Now, let's substitute the given values into the standard equation of a circle:
Center ( h , k ) = ( 5 , − 1 ) Radius r = 16
So, the equation becomes: ( x − 5 ) 2 + ( y − ( − 1 ) ) 2 = 1 6 2 ( x − 5 ) 2 + ( y + 1 ) 2 = 256
Compare with the options. Comparing our result with the given options: 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
We can see that option B matches our derived equation.
State the final answer. Therefore, the correct equation representing a circle with center T ( 5 , − 1 ) and a radius of 16 units is ( x − 5 ) 2 + ( y + 1 ) 2 = 256 .
Examples
Understanding the equation of a circle is crucial in various real-world applications. For instance, when designing a circular garden, you need to determine the layout based on the desired center point and radius. The equation of the circle helps you define the boundary of the garden, ensuring it fits perfectly within the available space. Similarly, in architecture and engineering, circular arches, domes, and other structural elements rely on the precise definition provided by the circle's equation to ensure stability and aesthetic appeal.
