What is the radius of a circle given by the equation $x^2+y^2-2 x+8 y-47=0$?
The given equation of the circle is in the general form \(x^2 + y^2 + Dx + Ey + F = 0\), where the equation is \(x^2 + y^2 - 2x + 8y - 47 = 0\). To find the radius, we need to convert this equation into 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 its radius.
To group the terms, rewrite the equation as \((x^2 - 2x) + (y^2 + 8y) = 47\). Now, we complete the square for both the \(x\) and \(y\) expressions.
For the \(x\) terms: \(x^2 - 2x = (x^2 - 2x + 1) - 1 = (x - 1)^2 - 1\).
For the \(y\) terms: \(y^2 + 8y = (y^2 + 8y + 16) - 16 = (y + 4)^2 - 16\).
Substituting these back into our equation, we get:
\((x - 1)^2 - 1 + (y + 4)^2 - 16 = 47\).
Moving the constants to the right side:
\((x - 1)^2 + (y + 4)^2 = 47 + 1 + 16\).
\((x - 1)^2 + (y + 4)^2 = 64\).
Comparing this to the standard form \((x - h)^2 + (y - k)^2 = r^2\), we can see that \(r^2 = 64\). Taking the square root of both sides gives \(r = \sqrt{64} = 8\). The radius of the circle is \(\boxed{8}\) units.
Understanding how to find the radius of a circle from its general equation is useful in fields like architecture and engineering, where circles are often defined by algebraic constraints. For example, if you are designing a circular fountain and its edge is defined by a coordinate equation in a plan, converting this to the standard form allows you to easily identify the radius and center, ensuring accurate placement and sizing in your design.