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)
Recall the standard equation of a circle: ( x − h ) 2 + ( y − k ) 2 = r 2 .
Compare the given equation with the standard equation to identify r 2 = 4 2 .
Take the square root to find the radius: r = 4 2 = 4 .
The radius of the circle is 4 units .
Explanation
Recall the standard equation of a circle The equation of a circle is given by ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center of the circle and r is the radius. In our case, the equation is ( x + 5 ) 2 + ( y − 3 ) 2 = 4 2 .
Compare with the standard equation Comparing the given equation ( x + 5 ) 2 + ( y − 3 ) 2 = 4 2 with the standard form ( x − h ) 2 + ( y − k ) 2 = r 2 , we can identify that r 2 = 4 2 .
Find the radius To find the radius r , we take the square root of both sides of the equation r 2 = 4 2 . Thus, r = 4 2 = 4 .
Examples
Understanding the equation of a circle helps in various real-world applications, such as designing circular gardens or determining the coverage area of a sprinkler. For instance, if you want to design a circular garden with a radius of 4 meters, you can use the equation ( x − h ) 2 + ( y − k ) 2 = 4 2 to map out the garden's boundaries on a coordinate plane, where ( h , k ) is the center of the garden. This ensures accurate planning and efficient use of space.
