Study Sheet of Conic Sections

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 = r2 where the center is (h,k) and the radius is r.

Alternate Form: x2 + y2 + 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:  Ax2 + 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:  Ay2 + 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: (x2/b2) + (y2/a2) = 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: (x2/a2) + (y2/b2) = 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: (y2/a2) – (x2/b2) = 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: (x2/a2) – (y2/b2) = 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).