What is the radius of a circle given by the equation $x^2+y^2-2 x+8 y-47=0$?
radius = $\square$ units
Answer (1)
Rewrite the given equation x 2 + y 2 − 2 x + 8 y − 47 = 0 by completing the square for both x and y terms. This involves expressing x 2 − 2 x as ( x − 1 ) 2 − 1 and y 2 + 8 y as ( y + 4 ) 2 − 16 .
Substitute these expressions back into the original equation and simplify to get the standard form of the circle equation: ( x − 1 ) 2 + ( y + 4 ) 2 = 64 .
Recognize that the equation is now in the form ( x − h ) 2 + ( y − k ) 2 = r 2 , where r 2 = 64 .
Determine the radius r by taking the square root of 64: r = 64 = 8 . Therefore, the radius of the circle is 8 units.
Explanation
Analyze the problem and rewrite the equation We are given the equation of a circle: x 2 + y 2 − 2 x + 8 y − 47 = 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 − 2 x . To complete the square, we take half of the coefficient of the x term, which is − 2 , so half of it is − 1 . Then we square it: ( − 1 ) 2 = 1 . So, we can rewrite x 2 − 2 x as ( x − 1 ) 2 − 1 .
Complete the square for y terms Next, we complete the square for the y terms. We have y 2 + 8 y . To complete the square, we take half of the coefficient of the y term, which is 8 , so half of it is 4 . Then we square it: ( 4 ) 2 = 16 . So, we can rewrite y 2 + 8 y as ( y + 4 ) 2 − 16 .
Substitute back into the original equation Now, we substitute these back into the original equation: ( x − 1 ) 2 − 1 + ( y + 4 ) 2 − 16 − 47 = 0 .
Simplify the equation We simplify the equation: ( x − 1 ) 2 + ( y + 4 ) 2 = 47 + 1 + 16 , which simplifies to ( x − 1 ) 2 + ( y + 4 ) 2 = 64 .
Identify the radius Now the equation is in the standard form ( x − 1 ) 2 + ( y + 4 ) 2 = 64 . We can identify the radius by taking the square root of the constant term on the right side: $r =
64 = 8 .
Therefore, the radius of the circle is 8 units.
State the final answer The radius of the circle is 8 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 a radius of 8 units (e.g., 8 meters), you can calculate the circumference to determine how much fencing to purchase. The circumference C is given by C = 2 π r , so in this case, C = 2 π ( 8 ) = 16 π ≈ 50.27 meters of fencing would be needed. This ensures you buy the correct amount of materials for your project.
