Question

what is the center and radius for a circle with the equation: (x - 4)^2 + (y + 2)^2 = 256

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 \((x - 4)^2+(y+2)^2 = 256\), we have \(h = 4\), \(k=- 2\) (since \(y+2=y-(-2)\)), and \(r^2 = 256\).

Step3: Calculate the radius

Take the square - root of \(r^2\). Since \(r=\sqrt{r^2}\) and \(r^2 = 256\), then \(r=\sqrt{256}=16\).

Answer:

Center: \((4,-2)\) Radius \( = 16\) units (the last option)