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)
The radius of the circle given by the equation ( x + 5 ) 2 + ( y − 3 ) 2 = 4 2 is 4 units, which corresponds to option B. This is derived from the comparison to the standard form of a circle's equation where r 2 = 16 . Thus, the radius is found by taking the square root of 16, resulting in 4.
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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: 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 is ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center of the circle and r is the radius.
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, architects and engineers use the equation of a circle to design circular structures such as domes and tunnels. Imagine you're designing a circular garden. Knowing the radius helps you determine the amount of fencing you need or the area you'll need to cover with soil. The equation of a circle allows for precise planning and construction of these circular elements.
