Determine the equation of each circle. Assume the radius is...

Key Concepts

Distance Formula

The distance formula, derived from the Pythagorean theorem, is used to calculate the distance between two points in the coordinate plane. This formula is essential in problems involving circles, especially when determining the radius from the distance between the center of the circle and a point on the circle. It also plays a key role in deriving and verifying the standard equation of a circle.

Radius

The radius is the constant distance from the center of the circle to any point on its circumference. In the standard equation, this distance is represented by r, and since the radius squared (r²) appears in the equation, it directly affects the size of the circle. In problems where the radius is assumed to be a whole number, it simplifies the calculation and interpretation of the circle’s dimensions.

Center Coordinates

The center of a circle is the fixed point from which every point on the circle is equidistant. In the equation of a circle, (h, k) are the center coordinates. Identifying or determining the center is crucial when forming the equation of a circle because it directly influences both the position and the shape of the circle in the coordinate plane.

Standard Equation of a Circle

This concept refers to the common form of the circle's equation, written as (x - h)² + (y - k)² = r². Here, (h, k) represents the coordinates of the circle’s center and r is the radius. Understanding this equation is fundamental to working with circles in coordinate geometry because it encapsulates the definition of a circle as all points equidistant from a fixed center.

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