What is the radius of a circle given by the equation $x^2+y^2-2 x+8 y-47=0 ?$
Answer (1)
Rewrite the given equation x 2 + y 2 − 2 x + 8 y − 47 = 0 by grouping the x and y terms: ( x 2 − 2 x ) + ( y 2 + 8 y ) − 47 = 0 .
Complete the square for the x terms: x 2 − 2 x = ( x − 1 ) 2 − 1 .
Complete the square for the y terms: y 2 + 8 y = ( y + 4 ) 2 − 16 .
Substitute these back into the original equation and simplify to get the standard form: ( x − 1 ) 2 + ( y + 4 ) 2 = 64 , so the radius is 8 .
Explanation
Analyze the problem and the given 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.
Group x and y terms To rewrite the given equation in standard form, we need to complete the square for both the x and y terms. First, let's group the x terms and y terms together: ( x 2 − 2 x ) + ( y 2 + 8 y ) − 47 = 0 .
Complete the square for x terms Now, let's 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 add and subtract 1 to complete the square: x 2 − 2 x + 1 − 1 = ( x − 1 ) 2 − 1 .
Complete the square for y terms Next, let's 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 add and subtract 16 to complete the square: y 2 + 8 y + 16 − 16 = ( y + 4 ) 2 − 16 .
Substitute back into the original equation Now, substitute these back into the original equation: (( x − 1 ) 2 − 1 ) + (( y + 4 ) 2 − 16 ) − 47 = 0 .
Simplify the equation Simplify the equation: ( x − 1 ) 2 − 1 + ( y + 4 ) 2 − 16 − 47 = 0 . ( x − 1 ) 2 + ( y + 4 ) 2 = 1 + 16 + 47 . ( x − 1 ) 2 + ( y + 4 ) 2 = 64 .
Identify r^2 Now the equation is in the standard form ( x − 1 ) 2 + ( y + 4 ) 2 = 64 . We can see that r 2 = 64 .
Find the radius To find the radius r , we take the square root of 64: $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. Similarly, in engineering, calculating the radius of a circular pipe is essential for determining its flow capacity. The ability to find the radius from the circle's equation allows for precise planning and execution in various projects, ensuring optimal use of space and resources.
