Write the standard form of the equation of the circle with the given center and radius. Center (3, 2), r = 5

Verified step by step guidance

1. Recall that the standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is given by the formula: \[ (x - h)^2 + (y - k)^2 = r^2 \]
2. Identify the center \((h, k)\) and the radius \(r\) from the problem. Here, the center is \((3, 2)\) and the radius is \(5\).
3. Substitute the values of \(h = 3\), \(k = 2\), and \(r = 5\) into the standard form equation: \[ (x - 3)^2 + (y - 2)^2 = 5^2 \]
4. Simplify the right side by squaring the radius: \[ (x - 3)^2 + (y - 2)^2 = 25 \]
5. Write the final equation, which represents the circle in standard form with the given center and radius.

Key Concepts

Standard Form of a Circle's Equation

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. This form directly shows the circle's center coordinates and radius, making it easy to graph and analyze.

Coordinates of the Center

The center of a circle is the fixed point equidistant from all points on the circle. In the equation, the center (h, k) shifts the circle horizontally and vertically on the coordinate plane, affecting the terms inside the parentheses.

Radius and Its Role in the Equation

The radius r is the distance from the center to any point on the circle. Squaring the radius (r²) in the equation determines the size of the circle, controlling how far the circle extends from its center.