Question

identify the center and radius of the circle written in standard form.
\\(x + 12)^2 + (y + 9)^2 = 144\\
center \\((x, y) = \\)
radius

Explanation:

Identify the standard equation of a circle

The standard equation of a circle is:

$$(x - h)^2 + (y - k)^2 = r^2$$

where \((h, k)\) is the center and \(r\) is the radius.

Match the given equation to standard form

The given equation is:

$$(x + 12)^2 + (y + 9)^2 = 144$$

We rewrite this to match the subtraction signs:

$$(x - (-12))^2 + (y - (-9))^2 = 144$$

Determine the center coordinates

Comparing the terms, we find:

$$h = -12$$

$$k = -9$$

Thus, the center \((h, k)\) is \((-12, -9)\).

Calculate the radius

The right side of the equation represents \(r^2\):

$$r^2 = 144$$

Taking the positive square root for a physical distance:

$$r = \sqrt{144} = 12$$

Answer:

Identify the center and radius of the circle written in standard form.

$$(x + 12)^2 + (y + 9)^2 = 144$$

center \((x, y) = \) <blank>\((-12, -9)\)</blank>
radius <blank>12</blank>