How are circles represented by equations on a coordinate plane?

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Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated. — analytic geometry The conic sections result from intersecting a plane with a double cone, as shown in the figure. There are three distinct families of conic sections: the ellipse (including the circle), the parabola (with one branch), and the hyperbola (with two branches).

In analytic geometry, a circle is defined as the set of all points $(x, y)$ that are a fixed distance $r$ (the radius) from a central point $(h, k)$ .

Using the Pythagorean theorem to express the distance between the center and any point on the curve, the standard form of the equation is:

$(x - h)^{2} + (y - k)^{2} = r^{2}$

In this equation:
* $(h, k)$ represents the coordinates of the center.
* $r$ represents the radius, where $r > 0$ .

If the center of the circle is located at the origin $(0, 0)$ , the equation simplifies to:

$x^{2} + y^{2} = r^{2}$

This algebraic representation allows for the precise study of circles within the coordinate plane, facilitating the calculation of intersections, tangents, and other geometric properties.1