Question

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

\\(x - 4)^2 + y^2 = \frac{16}{49}\\

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

radius \\(\quad\\)

Explanation:

🆕 New Concept Discovered: Equation of a Circle
Finding the center and radius from standard form

Step 1: Identify the standard form equation

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

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

The given equation is:

$$ (x - 4)^2 + y^2 = \frac{16}{49} $$

We can rewrite \( y^2 \) as \( (y - 0)^2 \) to match the standard form:

$$ (x - 4)^2 + (y - 0)^2 = \frac{16}{49} $$

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

By comparing the terms:

• \( x - h = x - 4 \implies h = 4 \)
• \( y - k = y - 0 \implies k = 0 \)

So, the center of the circle is:

$$ (x, y) = (4, 0) $$

Step 3: Find the radius \(r\)

By comparing the constant on the right side:

$$ r^2 = \frac{16}{49} $$

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

$$ r = \sqrt{\frac{16}{49}} = \frac{\sqrt{16}}{\sqrt{49}} = \frac{4}{7} $$

Answer:

• center \((x, y) =\) (4, 0)
• radius \(=\) 4/7