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 ( x − 1 ) 2 + ( y + 4 ) 2 − 1 − 16 − 47 = 0 .
Further simplify the equation to the standard form of a circle: ( x − 1 ) 2 + ( y + 4 ) 2 = 64 .
Identify the radius by recognizing that r 2 = 64 , so the radius is 8 units.
Explanation
Analyze the problem and state the goal. 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 and y terms. To rewrite the given equation in standard form, we need to complete the square for both the x and y terms.
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), square it ( ( − 1 ) 2 = 1 ), and add it to the expression. So, x 2 − 2 x + 1 = ( x − 1 ) 2 . Thus, x 2 − 2 x = ( x − 1 ) 2 − 1 .
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), square it ( 4 2 = 16 ), and add it to the expression. So, y 2 + 8 y + 16 = ( y + 4 ) 2 . Thus, y 2 + 8 y = ( y + 4 ) 2 − 16 .
Substitute back into the original equation. Now, substitute these expressions back into the original equation: ( x − 1 ) 2 − 1 + ( y + 4 ) 2 − 16 − 47 = 0 .
Simplify the equation. Simplify the equation by combining the constant terms: ( x − 1 ) 2 + ( y + 4 ) 2 − 1 − 16 − 47 = 0 ( x − 1 ) 2 + ( y + 4 ) 2 − 64 = 0 ( 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 . Comparing this to ( x − h ) 2 + ( y − k ) 2 = r 2 , we see that r 2 = 64 . To find the radius r , we take the square root of 64: $r = 64 = 8
State the final answer. Therefore, the radius of the circle is 8 units.
Examples
Understanding the radius of a circle is crucial in many real-world applications. For instance, 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 convert a circle's equation into standard form and extract its radius is a fundamental skill in geometry and has practical implications in various fields.
