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 = ( x + 4 ) 2 − 16 .
Complete the square for the y terms: 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: 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 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 Simplify the equation: ( 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 Now we have the equation 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.
State the final answer The radius of the circle is 2 units.
Examples
Understanding the equation of a circle is crucial 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. Similarly, in physics, understanding circular motion relies on knowing the radius of the circular path. This concept also extends to fields like astronomy, where the orbits of planets are approximated as circles, and the radius helps calculate orbital parameters.
