What is the radius of a circle whose equation is $(x+5)^2+(y-3)^2=4^2$?
A. 2 units
B. 4 units
C. 8 units
D. 16 units
Answer (2)
Recognize the standard form of a circle's equation: ( x − h ) 2 + ( y − k ) 2 = r 2 .
Compare the given equation ( x + 5 ) 2 + ( y − 3 ) 2 = 4 2 to the standard form.
Identify that r 2 = 4 2 .
Determine the radius by taking the square root: r = 4 .
Explanation
Analyze the problem The equation of a circle is given as ( x + 5 ) 2 + ( y − 3 ) 2 = 4 2 . We need to find the radius of the circle.
Recall the standard equation of a circle The standard equation of a circle with center ( h , k ) and radius r is given by ( x − h ) 2 + ( y − k ) 2 = r 2 .
Compare the given equation with the standard equation Comparing the given equation ( x + 5 ) 2 + ( y − 3 ) 2 = 4 2 with the standard equation, we can see that r 2 = 4 2 .
Find the radius Taking the square root of both sides of r 2 = 4 2 , we get r = 4 . Therefore, the radius of the circle is 4 units.
State the final answer The radius of the circle is 4 units.
Examples
Understanding the equation of a circle is useful in many real-world applications. For example, when designing a circular garden, you need to know the radius to determine how much fencing you'll need. Similarly, in architecture, circular windows or domes require precise calculations of the radius to ensure structural integrity and aesthetic appeal. In sports, the radius of a circular track is crucial for calculating distances and optimizing performance. The equation of a circle, ( x − h ) 2 + ( y − k ) 2 = r 2 , helps us relate the coordinates of any point on the circle to its center and radius, enabling us to solve various practical problems.
The radius of the circle given by the equation ( x + 5 ) 2 + ( y − 3 ) 2 = 4 2 is 4 units, as determined by taking the square root of 4 2 . The correct choice is B. 4 units.
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