The graph of [tex]4 x^2+4 y^2-24 x-32 y+72=0[/tex] is a circle. What is the radius of the circle?
Answer (2)
Divide the given equation by 4 to simplify it.
Complete the square for both x and y terms.
Rewrite the equation in the standard form of a circle: ( x − 3 ) 2 + ( y − 4 ) 2 = 7 .
Identify the radius as the square root of the constant term: 7 .
Explanation
Analyze the problem and the given equation We are given the equation of a circle: 4 x 2 + 4 y 2 − 24 x − 32 y + 72 = 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.
Simplify the equation First, divide the entire equation by 4 to simplify it: 4 4 x 2 + 4 y 2 − 24 x − 32 y + 72 = 4 0 x 2 + y 2 − 6 x − 8 y + 18 = 0
Group x and y terms Next, rearrange the terms to group the x and y terms together: ( x 2 − 6 x ) + ( y 2 − 8 y ) = − 18
Complete the square Now, complete the square for the x terms. To do this, take half of the coefficient of the x term (-6), square it, and add it to both sides. Half of -6 is -3, and ( − 3 ) 2 = 9 .
Similarly, complete the square for the y terms. Take half of the coefficient of the y term (-8), square it, and add it to both sides. Half of -8 is -4, and ( − 4 ) 2 = 16 .
So, we add 9 and 16 to both sides of the equation: ( x 2 − 6 x + 9 ) + ( y 2 − 8 y + 16 ) = − 18 + 9 + 16
Rewrite in standard form Rewrite the equation in the standard form: ( x − 3 ) 2 + ( y − 4 ) 2 = − 18 + 9 + 16 ( x − 3 ) 2 + ( y − 4 ) 2 = 7
Identify the radius Now, we can identify the radius r from the standard equation ( x − 3 ) 2 + ( y − 4 ) 2 = 7 . Since r 2 = 7 , we have r = 7 .
State the final answer The radius of the circle is 7 .
Examples
Understanding the equation of a circle is crucial in various fields, such as engineering and computer graphics. For instance, when designing a circular garden or a roundabout, knowing the radius helps determine the amount of fencing or paving material needed. In computer graphics, circles are fundamental elements in creating images and animations, and their equations are used to define their position and size.
The radius of the circle described by the equation 4 x 2 + 4 y 2 − 24 x − 32 y + 72 = 0 is 7 . This value is derived after simplifying the equation and rewriting it in standard form. By completing the square for both the x and y terms, we determine the radius as the square root of the constant term on the right side.
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