Question

find the center and the radius of the circle. then graph the circle.
( x^2 = 25 - (y + 3)^2 )
use the graphing tool to graph the circle.
click to enlarge graph

Explanation:

Step1: Rewrite the equation in standard form

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. Start with the given equation \(x^2 = 25 - (y + 3)^2\). Move \((y + 3)^2\) to the left side:
\(x^2 + (y + 3)^2 = 25\)

Step2: Identify \(h\), \(k\), and \(r\)

Compare \(x^2 + (y + 3)^2 = 25\) with \((x - h)^2 + (y - k)^2 = r^2\).

• For the \(x\)-term: \(x^2=(x - 0)^2\), so \(h = 0\).
• For the \(y\)-term: \((y + 3)^2=(y - (-3))^2\), so \(k=-3\).
• For the radius: \(r^2 = 25\), so \(r=\sqrt{25}=5\) (since radius is positive).

Answer:

The center of the circle is \((0, -3)\) and the radius is \(5\).