Definition of a circle, parabola, ellipses, hyperbola and their conic sections, equation and standard form.

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A circle is one of the fundamental conic sections. It is defined as the set of all points in a plane that are equidistant from a fixed point called the center. The standard form of a circle's equation is (x minus h) squared plus (y minus k) squared equals r squared, where (h, k) represents the center coordinates and r is the radius. A parabola is another fundamental conic section. It is defined as the set of all points in a plane that are equidistant from a fixed point called the focus and a fixed line called the directrix. The standard forms are (x minus h) squared equals 4p times (y minus k) for vertical parabolas, and (y minus k) squared equals 4p times (x minus h) for horizontal parabolas, where (h, k) is the vertex and p determines the direction and width. An ellipse is defined as the set of all points in a plane such that the sum of distances from two fixed points called foci is constant. The standard forms are (x minus h) squared over a squared plus (y minus k) squared over b squared equals 1 for horizontal major axis, and (x minus h) squared over b squared plus (y minus k) squared over a squared equals 1 for vertical major axis, where (h, k) is the center and a and b are the semi-major and semi-minor axes. A hyperbola is defined as the set of all points in a plane such that the absolute difference of distances from two fixed points called foci is constant. The standard forms are (x minus h) squared over a squared minus (y minus k) squared over b squared equals 1 for horizontal opening, and (y minus k) squared over a squared minus (x minus h) squared over b squared equals 1 for vertical opening, where (h, k) is the center. Hyperbolas have asymptotes that guide their shape. In summary, conic sections are fundamental curves in mathematics formed by intersecting a plane with a double cone. A circle forms when the plane is perpendicular to the cone's axis, an ellipse when the plane cuts at an angle, a parabola when the plane is parallel to a generator line, and a hyperbola when the plane cuts through both cones. All conic sections can be represented by the general equation Ax squared plus Bxy plus Cy squared plus Dx plus Ey plus F equals zero, where the discriminant B squared minus 4AC determines the type of conic.