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

Rewrite the given equation x 2 + y 2 + 8 x − 6 y + 21 = 0 by completing the square for both x and y terms.
Complete the square for x : x 2 + 8 x = ( x + 4 ) 2 − 16 .
Complete the square for y : y 2 − 6 y = ( y − 3 ) 2 − 9 .
Substitute these back into the original equation and simplify to get ( 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 as 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's equation, 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 the x term (which is 8), square it (which is ( 8/2 ) 2 = 4 2 = 16 ), and add and subtract it. So, x 2 + 8 x = ( x 2 + 8 x + 16 ) − 16 = ( 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 the y term (which is -6), square it (which is ( − 6/2 ) 2 = ( − 3 ) 2 = 9 ), and add and subtract it. So, y 2 − 6 y = ( y 2 − 6 y + 9 ) − 9 = ( 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 We simplify the equation by combining the constant terms: ( x + 4 ) 2 − 16 + ( y − 3 ) 2 − 9 + 21 = 0 ( 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 The equation is now in the standard form ( x + 4 ) 2 + ( y − 3 ) 2 = 4 . Comparing this to ( x − h ) 2 + ( y − k ) 2 = r 2 , we see that r 2 = 4 . Taking the square root of both sides, we get r = 4 ​ = 2 . Therefore, the radius of the circle is 2 units.


Examples
Understanding the equation of a circle is useful in various real-world applications. For instance, when designing a circular garden or a roundabout, knowing the radius helps determine the amount of fencing or paving material needed. Also, in fields like astronomy, the orbits of planets and satellites can be approximated as circles, and determining their radii is crucial for predicting their paths and positions.

Answered by GinnyAnswer | 2025-07-08