What is the radius of a circle whose equation is $x^2+y^2+8 x-6 y+21=0$?
A. 2 units
B. 3 units
C. 4 units
D. 5 units
Answer (1)
Rewrite the given equation x 2 + y 2 + 8 x − 6 y + 21 = 0 in the standard form of a circle's equation.
Complete the square for the x terms: x 2 + 8 x becomes ( x + 4 ) 2 − 16 .
Complete the square for the y terms: y 2 − 6 y becomes ( y − 3 ) 2 − 9 .
The equation simplifies to ( x + 4 ) 2 + ( y − 3 ) 2 = 4 , so the radius is 2 .
Explanation
Analyze the problem and rewrite in standard form We are given the equation of a circle: x 2 + y 2 + 8 x − 6 y + 21 = 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 + 8 x . To complete the square, we take half of the coefficient of x , which is 2 8 = 4 , and square it, which is 4 2 = 16 . So, we can rewrite x 2 + 8 x as ( x + 4 ) 2 − 16 .
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 y , which is 2 − 6 = − 3 , and square it, which is ( − 3 ) 2 = 9 . So, we can rewrite y 2 − 6 y as ( y − 3 ) 2 − 9 .
Substitute back into original equation Now, we substitute these back into the original equation: x 2 + y 2 + 8 x − 6 y + 21 = 0 becomes ( x + 4 ) 2 − 16 + ( y − 3 ) 2 − 9 + 21 = 0 .
Simplify the equation Simplify the equation by combining the constants: ( x + 4 ) 2 + ( y − 3 ) 2 − 16 − 9 + 21 = 0 ( x + 4 ) 2 + ( y − 3 ) 2 − 4 = 0 ( x + 4 ) 2 + ( y − 3 ) 2 = 4 .
Identify the radius Now the equation is in the standard form ( x + 4 ) 2 + ( y − 3 ) 2 = 4 . We can see that r 2 = 4 . To find the radius r , we take the square root of 4: r = 4 = 2 .
State the final answer Therefore, the radius of the circle is 2 units.
Examples
Understanding the equation of a circle is crucial in various fields. For instance, in GPS technology, your location is determined by finding the intersection of circles from multiple satellites. Each satellite sends a signal indicating your distance from it, which defines a circle (or sphere in 3D). The radius we calculated is similar to finding the range of these signals, helping to pinpoint exact locations on Earth.
