Question

find the center and the radius of the circle.
$(x - 4)^2 + (y + 5)^2 = 73$
the center is $square$.
(type an ordered pair.)
the radius is $square$.
(type an exact answer, using radicals as needed.)

Explanation:

Step1: Recall the standard circle equation

The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Step2: Identify \(h\), \(k\), and \(r\) from the given equation

For the equation \((x - 4)^2 + (y + 5)^2 = 73\), we can rewrite \(y + 5\) as \(y - (-5)\). Comparing with the standard form:

• \(h = 4\) (from \(x - 4\)), \(k = -5\) (from \(y - (-5)\)), so the center is \((4, -5)\).
• For the radius, we have \(r^2 = 73\), so \(r = \sqrt{73}\) (since radius is positive).

Answer:

The center is \((4, -5)\).
The radius is \(\sqrt{73}\).