Conic sections

Any one of the shapes that can be created when a plane and a double-napped cone intersect are called conic sections, or conics. There are four kinds of conic sections: circles, parabolas, ellipses, and hyperbolas. In all standard form equations, the point $$ (h\,\!,k\,\!)$$ is the center

Circles
A circle is defined as a closed plane curve consisting of the set of all points in a plane equidistant to a point called the center.

The equation of a circle is: $$(x - h)^2 + (y-k)^2 = r^2$$, where $$r$$ is the radius.

Parabolas
A parabola is defined as the set of all points in a plane equidistant from a fixed line AND a fixed point.

The standard form for writing a parabola with a horizontal axis of symmetry is $$ (y-k)^2= 4p(x-h)\,\!$$ while a parabola with a vertical axis of symmetry is $$ (x-h)^2= 4p(y-k)\,\!$$.

Ellipses
An ellipse is the set of all points in a plane where the sum of distances between the points and two foci are a constant. The standard equation for an ellipse is $$(x - h)^2/a^2 + (y - k)^2/b^2 = 1$$. If $$a$$ and $$b$$ are equal, the shape is a circle.

Hyperbolas
A hyperbola is the set of all points in a plane where the difference of distances between the points and two foci are a constant. It is formed when a plane intersects a double-napped cone through the center.