The circle $C$ has equation:
$x^2 + y^2 - 10x + 6y + 30 = 0$
Find the radius of $C$.
The equation of the circle is given by \(x^2 + y^2 - 10x + 6y + 30 = 0\). To find the radius, we need to rewrite this in the standard form of a circle's equation, which is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) represents the center and \(r\) represents the radius. We begin by grouping the \(x\) and \(y\) terms:
\((x^2 - 10x) + (y^2 + 6y) = -30\).
To complete the square for the \(x\) terms, we take half of the coefficient of \(x\), which is \(-10/2 = -5\), and square it to get \(25\). To complete the square for the \(y\) terms, we take half of the coefficient of \(y\), which is \(6/2 = 3\), and square it to get \(9\). We then add these values to both sides of the equation:
\((x^2 - 10x + 25) + (y^2 + 6y + 9) = -30 + 25 + 9\).
Now we simplify both sides of the equation. On the left side, the expressions are perfect squares. On the right side, we perform the arithmetic:
\((x - 5)^2 + (y + 3)^2 = 4\).
Comparing this to the standard form \((x - h)^2 + (y - k)^2 = r^2\), we can see that \(r^2 = 4\). Thus, the radius \(r\) is \(\sqrt{4} = 2\). The final answer is \(\boxed{2}\).
In architectural design, architects often represent circular architectural features, such as circular windows or pillars, on a coordinate grid to calculate their dimensions. If a window's boundary is defined by the equation \(x^2 + y^2 - 10x + 6y + 30 = 0\), finding the radius is essential for determining how much glass is needed or how much space the feature will occupy on a wall. By completing the square, the architect can quickly identify the center at \((5, -3)\) and the radius as \(2\) units, ensuring precise planning for installation.