## Definitions

- The
**Alternate Form**of the equation of a conic section is different for each type. Generally, though, the alternate form takes the Standard Form and sets it equal to zero. **Asymptotes**are two straight lines that intersect at the center of a hyperbola that help define its shape and size.- The
**Axis of Symmetry**is the straight line that passes through the focus and vertex of a parabola - The
**Center**is the exact middle of all conic sections except the parabola. It is the intersection of the major axis and minor axis, and also the intersection of the asymptotes in a hyperbola. - A
**Circle**is the set of all points that are equally distant from a fixed point, otherwise known as the center. - A
**Conic Section**is a curve formed by the intersection of a cone with a plane. Examples of conic sections include the circle, parabola, ellipse, and hyperbola. - The
**Directrix**of a parabola is a straight line perpendicular to the axis of symmetry. - An
**Ellipse**is the set of all points surrounding two foci or focus points, such that the sum of the distances from any point on the ellipse to each focus point remains constant. - The
**Focus**, or**Foci**(plural) is one or two points that lie along the major axis of a conic section and define its shape. - A
**Hyperbola**is the set of all points around two foci, or focus points, such that the difference of the distances from any point on the hyperbola to each focus point is a positive constant. - The
**Latus Rectum**is a straight line drawn through a focus point that is perpendicular to the major axis in all conic sections except the circle. In the parabola, the latus rectum is parallel to the directrix. - In the ellipse and hyperbola, the
**Major Axis**is the straight line drawn through the two focus points, connecting one side of the conic section to the other. - In the ellipse and hyperbola, the
**Minor Axis**is the straight line drawn through the center of the conic section, perpendicular to the major axis. - A
**Parabola**is the set of points that are equally distant from both a fixed point (the focus) and a fixed line (the directrix). - The
**Vertex**is the intersection of the axis of symmetry and the parabola. In the ellipse and hyperbola, the**Vertices**(plural) are the endpoints of the major axis.

## Equation of a Circle

Standard Form:**(x – h) ^{2} + (y – k)^{2 }= r**

^{2 }where the center is (h,k) and the radius is r.

Alternate Form: **x ^{2} + y^{2} + Dx + Ey + F = 0** where D, E, and F are real numbers.

## Equation of a Vertical Parabola

Standard Form: **(x – h) ^{2} = 4p(y – k)**

^{ }where the vertex is at (h, k). The parabola opens upward if p>0 and downward if p<0.

Alternate Form: **Ax ^{2} + Bx +Cy + D = 0**

^{ }where A, B, C, and D are integers and A and C are non-zero.

## Equation of a Horizontal Parabola

Standard Form: **(y – k) ^{2} = 4p(x – h)**

^{ }where the vertex is at (h, k). The parabola opens to the left if p>0 and to the right if p<0.

Alternate Form: **Ay ^{2} + By +Cx + D = 0**

^{ }where A, B, C, and D are integers and A and C are non-zero.

## Equation of a Vertical Ellipse

Standard Form:

**(x**^{2}/b^{2}**) +****(y**^{2}/a^{2}**) = 1****where a is half the length of the major axis, b is half the length of the minor axis, and the ellipse is centered on (0,0).**## Equation of a Horizontal Ellipse

Standard Form:

**(x**^{2}/a^{2}**) +****(y**^{2}/b^{2}**) = 1****where a is half the length of the major axis, b is half the length of the minor axis, and the ellipse is centered on (0,0).**## Equation of a Vertical Hyperbola

Standard Form:

**(y**^{2}/a^{2}**) –****(x**^{2}/b^{2}**) = 1****where a is half the length of the major axis, b is half the length of the minor axis, and the hyperbola is centered on (0,0).**## Equation of a Horizontal Hyperbola

Standard Form:

**(x**^{2}/a^{2}**) –****(y**^{2}/b^{2}**) = 1****where a is half the length of the major axis, b is half the length of the minor axis, and the hyperbola is centered on (0,0).**