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 circle equation in standard form by completing the square for both x and y terms.
Express x 2 + 8 x as ( x + 4 ) 2 − 16 and y 2 − 6 y as ( y − 3 ) 2 − 9 .
Substitute these back into the original equation and simplify to get ( x + 4 ) 2 + ( y − 3 ) 2 = 4 .
Identify the radius from the standard form, where r 2 = 4 , thus r = 4 = 2 . The radius is 2 .
Explanation
Analyze the problem and the given equation 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 the original equation Now, we substitute these back into the original equation: ( x + 4 ) 2 − 16 + ( y − 3 ) 2 − 9 + 21 = 0 .
Simplify the equation We simplify the equation by combining the constants: ( x + 4 ) 2 + ( y − 3 ) 2 − 16 − 9 + 21 = 0 . This simplifies to ( x + 4 ) 2 + ( y − 3 ) 2 − 4 = 0 .
Isolate the constant term We isolate the constant term on the right side of the equation: ( x + 4 ) 2 + ( y − 3 ) 2 = 4 .
Identify the radius Now, we can identify the radius by comparing this to the standard form ( x − h ) 2 + ( y − k ) 2 = r 2 . We have r 2 = 4 , so r = 4 = 2 . Therefore, the radius of the circle is 2 units.
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 an equation similar to the one we solved, knowing how to find the radius allows you to plan the layout and purchase the correct amount of materials. This concept is also used in architecture, engineering, and even in creating art, where circular shapes and their dimensions play a significant role.
