Write equations of circles in standard form

Learning Objectives Define the standard form of a circle's equation and its components. Identify the center (h, k) and radius (r) from a given equation in standard form. Write the standard form equation of a circle given its center and radius. Derive the standard form equation of a circle given its center and a point on the circle. Determine the standard form equation of a circle given the endpoints of its diameter. Write the equation of a circle given its center and its tangency to a horizontal or vertical line. Ever wondered how GPS pinpoints your location? 📍 It often involves the intersection of signals, which can be modeled as circles. Let's learn the fundamental language of these shapes! In this tutorial, you will master writing the equation of a circle in i...

TermDefinitionExample CircleThe set of all points in a plane that are equidistant from a single fixed point, called the center.All the points on the edge of a wheel are equidistant from the axle in the middle. Center (h, k)The fixed point from which all points on the circle are the same distance. It is represented by the coordinates (h, k).For a circle with its center at the origin, (h, k) is (0, 0). Radius (r)The fixed distance from the center to any point on the circle. The radius must be a positive number.If the center is at (2, 3) and a point on the circle is (2, 8), the radius is 5 units. Standard FormThe equation of a circle written as (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.The equation (x - 1)² + (y + 4)² = 9 represents a circle with its center at...

Standard Form of a Circle (x - h)² + (y - k)² = r² This is the fundamental equation for a circle. The variables h and k represent the x and y coordinates of the center, and r represents the radius. Note that the radius is squared in the equation. Distance Formula (for finding radius) r = √((x₂ - x₁)² + (y₂ - y₁)²) Use this formula to find the radius (r) when you are given the center (x₁, y₁) and a point on the circle (x₂, y₂). The radius is the distance between these two points. Midpoint Formula (for finding center) (h, k) = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 ) Use this formula to find the center of a circle (h, k) when you are given the two endpoints of a diameter, (x₁, y₁) and (x₂, y₂). The center is the midpoint of the diameter.