Question

identify the center and radius of the circle written in standard form.
\\x^2 + (y + 4)^2 = 8\\

center \\((x, y) = (\quad)\\)
radius \\(\quad\\)

Explanation:

🆕 New Concept Discovered: Equation of a Circle
Standard form reveals the center and radius

Step 1: Identify the standard form equation

The standard form of the equation of a circle with center \( (h, k) \) and radius \( r \) is:

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

We are given the equation:

$$ x^2 + (y + 4)^2 = 8 $$

Step 2: Find the center \((h, k)\)

Rewrite the given equation to match the standard form exactly:

$$ (x - 0)^2 + (y - (-4))^2 = 8 $$

By comparing the terms:

• \( h = 0 \)
• \( k = -4 \)

So, the center \( (x, y) \) is \( (0, -4) \).

Step 3: Find the radius \(r\)

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

$$ r^2 = 8 $$

To find the radius \( r \), take the positive square root of both sides:

$$ r = \sqrt{8} $$

Simplify the radical:

$$ r = \sqrt{4 \cdot 2} = 2\sqrt{2} $$

Answer:

• center \((x, y)\) = \((0, -4)\)
• radius = \(2\sqrt{2}\) (or \(\sqrt{8}\))