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 (1)
Recognize the equation of a circle is in the form ( x − h ) 2 + ( y − k ) 2 = r 2 .
Identify that r 2 = 4 2 = 16 .
Calculate the square root to find the radius: r = \[ \sqrt{16} \] = 4 .
Conclude that the radius of the circle is 4 units.
Explanation
Analyze the problem The equation of a circle is given by ( 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 form of the equation of a circle with center ( h , k ) and radius r is ( 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 ( x − h ) 2 + ( y − k ) 2 = r 2 , we can identify that r 2 = 4 2 = 16 .
Find the radius To find the radius r , we take the square root of r 2 :
r = \[ \sqrt{16} = 4 \]
State the final answer Therefore, the radius of the circle is 4 units.
Examples
Understanding the equation of a circle is crucial in various real-world applications. For instance, when designing a circular garden, knowing the radius helps determine the amount of fencing needed. Similarly, in architecture, circular windows or domes require precise radius calculations to ensure structural integrity and aesthetic appeal. In navigation, the radius of the Earth is used to calculate distances and positions. This concept also forms the basis for understanding planetary orbits and the movement of celestial bodies.
