Write the standard form of the equation of the circle with the given center and radius. Center (-1, 4), r = 2

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 radius \(r\) from the problem. Here, the center is \((-1, 4)\) and the radius is \(2\).
3. Substitute the values of \(h = -1\), \(k = 4\), and \(r = 2\) into the standard form equation: \[ (x - (-1))^2 + (y - 4)^2 = 2^2 \]
4. Simplify the expression inside the parentheses and the right side: \[ (x + 1)^2 + (y - 4)^2 = 4 \]
5. This equation 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 or analyze.

Coordinates of the Center

The center of a circle is a fixed point equidistant from all points on the circle. In the equation, the center (h, k) shifts the circle from the origin, affecting the signs inside the parentheses as (x - h) and (y - k).

Radius and Its Role in the Equation

The radius r is the distance from the center to any point on the circle. In the equation, r is squared (r²) and represents the constant sum of squared distances in x and y directions, defining the circle's size.