Write equations of circles in standard form using properties

Learning Objectives Identify the center and radius of a circle from its given properties. Recall and apply the standard form equation of a circle. Write the equation of a circle given its center and radius. Write the equation of a circle given its center and a point on the circle. Write the equation of a circle given the endpoints of a diameter. Determine the equation of a circle that is tangent to an axis. Translate graphical representations of circles into their standard form equations. Ever wonder how GPS pinpoints your location? 📍 It uses the intersection of circles! Let's learn the algebraic language of these perfect shapes. This tutorial will guide you through writing the standard form equation of a circle using its key properties like the center and radius. M...

TermDefinitionExample CircleThe set of all points in a plane that are equidistant from a fixed point called the center.All the points on the edge of a wheel are equidistant from the axle in the center. Center (h, k)The fixed point from which all points on the circle are equidistant. It is represented by the coordinates (h, k).For a circle centered at the origin, the center is (0, 0). For a circle shifted 3 units right and 2 units down, the center is (3, -2). 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 (1, 2) and a point on the circle is (1, 5), the radius is the distance between them, which is 3 units. Diameter (d)A line segment passing through the center of a circle with both endpoints on the circle. Its l...

Standard Form Equation of a Circle (x - h)^2 + (y - k)^2 = r^2 Use this formula to write the equation of a circle when you know the coordinates of the center (h, k) and the length of the radius (r). Distance Formula d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} Use this to find the distance between two points. It is essential for finding the radius (r) when given the center and a point on the circle, or for finding the length of the diameter. Midpoint Formula M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) Use this to find the center of a circle (h, k) when you are given the two endpoints of a diameter.