What is the radius of a circle whose equation is $x^2+y^2-10 x+6 y+18=0$?
A. 2 units
B. 4 units
C. 8 units
D. 16 units
Answer (1)
Complete the square for both x and y terms in the given equation.
Rewrite the equation in the standard form ( x − h ) 2 + ( y − k ) 2 = r 2 .
Identify the radius r by taking the square root of the constant term on the right side of the equation.
The radius of the circle is 4 units.
Explanation
Analyze the problem and the given data. We are given the equation of a circle: x 2 + y 2 − 10 x + 6 y + 18 = 0 . Our goal is to find the radius of this circle. To do this, we will rewrite the equation in the standard form of a circle, which is ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center of the circle and r is the radius.
Complete the square for x terms. First, we complete the square for the x terms. We have x 2 − 10 x . To complete the square, we take half of the coefficient of the x term, which is − 10/2 = − 5 , and square it, which is ( − 5 ) 2 = 25 . So, we can write x 2 − 10 x = ( x − 5 ) 2 − 25 .
Complete the square for y terms. Next, we complete the square for the y terms. We have y 2 + 6 y . To complete the square, we take half of the coefficient of the y term, which is 6/2 = 3 , and square it, which is ( 3 ) 2 = 9 . So, we can write y 2 + 6 y = ( y + 3 ) 2 − 9 .
Substitute back into the original equation. Now, we substitute these back into the original equation: ( x − 5 ) 2 − 25 + ( y + 3 ) 2 − 9 + 18 = 0 .
Simplify the equation. We simplify the equation by moving the constants to the right side: ( x − 5 ) 2 + ( y + 3 ) 2 = 25 + 9 − 18 .
Calculate the constant term. We calculate the right side of the equation: 25 + 9 − 18 = 34 − 18 = 16 . So, the equation becomes ( x − 5 ) 2 + ( y + 3 ) 2 = 16 .
Identify the radius. Finally, we identify the radius by taking the square root of the right side: $r =
16 = 4 . Therefore, the radius of the circle is 4.
Examples
Understanding the radius of a circle is crucial in many real-world applications. For example, when designing a circular garden, knowing the radius helps determine the amount of fencing needed. If you want a circular garden with the equation ( x − 2 ) 2 + ( y − 3 ) 2 = 9 , the radius is 9 = 3 units. This tells you how far from the center ( 2 , 3 ) the edge of the garden will extend, allowing you to plan your space effectively.
