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 the circle is in the form ( x − h ) 2 + ( y − k ) 2 = r 2 .
Identify that r 2 = 4 2 from the given equation.
Calculate the radius by taking the square root: r = 4 2 = 4 .
Conclude that the radius of the circle is 4 units.
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 identify that r 2 = 4 2 .
Find the radius Taking the square root of both sides of r 2 = 4 2 , we get r = 4 2 = 4 . Therefore, the radius of the circle is 4 units.
Examples
Understanding the equation of a circle is crucial in various fields, such as architecture and engineering. For instance, when designing a circular window or a cylindrical pillar, knowing the radius from the equation helps in determining the dimensions and ensuring structural integrity. Imagine you are designing a circular garden with the equation ( x + 5 ) 2 + ( y − 3 ) 2 = 4 2 . The radius, which is 4 units, tells you how far from the center the edge of the garden will extend, allowing you to plan the layout effectively.
