Question

the equation of a circle is given below. identify the radius and the center. then graph the circle.
$x^2 + y^2 + 8x - 6y = -21$
radius:
center: ( , )

Explanation:

Step1: Complete the square for x-terms

Given the equation \(x^2 + y^2 + 8x - 6y = -21\). For the \(x\)-terms, \(x^2 + 8x\), we take half of 8 (which is 4), square it (\(4^2 = 16\)) and add it to both sides. For the \(y\)-terms, \(y^2 - 6y\), half of -6 is -3, square it (\((-3)^2 = 9\)) and add it to both sides.
So, \(x^2 + 8x + 16 + y^2 - 6y + 9 = -21 + 16 + 9\)

Step2: Rewrite as perfect squares

The left side becomes \((x + 4)^2 + (y - 3)^2\), and the right side is \(-21 + 16 + 9 = 4\)
The standard form of a circle is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
Comparing, we have \(h = -4\), \(k = 3\), and \(r^2 = 4\), so \(r = \sqrt{4} = 2\)

Answer:

Radius: \(2\)
Center: \((-4, 3)\)