What is the radius of a circle whose equation is $x^2+y^2+8 x-6 y+21=0$?
Answer (1)
Rewrite the given equation x 2 + y 2 + 8 x − 6 y + 21 = 0 by completing the square for both x and y terms.
Express the equation in the standard form ( x − h ) 2 + ( y − k ) 2 = r 2 .
Identify r 2 from the standard equation and calculate the radius r .
The radius of the circle is 2 units.
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'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 Simplify the equation: ( x + 4 ) 2 − 16 + ( y − 3 ) 2 − 9 + 21 = 0 , which becomes ( x + 4 ) 2 + ( y − 3 ) 2 − 16 − 9 + 21 = 0 . Then, ( x + 4 ) 2 + ( y − 3 ) 2 − 4 = 0 .
Rewrite in standard form Rewrite the equation in the standard form: ( x + 4 ) 2 + ( y − 3 ) 2 = 4 .
Identify the radius Now we identify the radius. Comparing the equation to ( x − h ) 2 + ( y − k ) 2 = r 2 , we see that r 2 = 4 . Therefore, r = 4 = 2 .
State the final answer The radius of the circle is 2 units.
Examples
Understanding the equation of a circle is crucial in various fields. For instance, in architecture, determining the radius helps in designing circular structures like domes or arches. In navigation, the radius can represent the range of a signal from a circular antenna. Also, in computer graphics, circles are fundamental shapes, and knowing their radius is essential for rendering and manipulating them.
