Pre Cal

• Circle (conic section)


  formed when the plane is parallel to the cone's base
• Ellipse (conic section)


  formed when the plane is between the cone's base and its apex
• Parabola (conic section)


  formed when the plane is at the same angle as the cone's slant
• Hyperbola (conic section)


  formed when the plane intersects the cone between its apex and its base
• • DEGENERATE CONIC SECTION A degenerate conic is generated when a plane intersects the vertex of the

  cone
• three types of degenerate conics:The degenerate form of a circle or an ellipse is a singular point.

  • The degenerate form of a parabola is a line.

  • The degenerate form of a hyperbola is two intersecting lines.
• Conic section


  Particular class of curves which often times appear in nature and which have applications in other fields. The conic sections are the nondegenerate curves generated by the intersections of a plane with one or two nappes of a cone.
• Types of conic sections

  • Circle
  • Ellipse
  • Parabola
  • Hyperbola
• Circle


  • When a plane that is parallel to the base of the cone intersects it, a circle will be generated.
• Ellipse


  • When a plane intersects the cone at an angle to form a bounded curve, an ellipse will be generated.
• Parabola


  • When a plane intersects the cone at an angle to form an unbounded curve, a parabola will be generated.
• Hyperbola


  • When a plane intersects both cones to form two unbounded curves, a hyperbola will be generated.
• Degenerate conic sections


  • The degenerate form of a circle or an ellipse is a singular point.
  • The degenerate form of a parabola is a line.
  • The degenerate form of a hyperbola is two intersecting lines.
• Circle


  The set of all points in a plane such that each point in the set is equidistant from a fixed point called the center. The distance from the center is called the radius. The distance around the circle is called the circumference.
• Standard equation of circle with center at (0,0)

  x^2 + y^2 = r^2
• Standard equation of circle with center at (h,k)


  (x - h)^2 + (y - k)^2 = r^2
• Parabola


  A locus of any point which is equidistant from a given point (focus) and a given line (directrix) is called a parabola.
• Parts of a parabola

  • Vertex (V) - The Minimum Point (Lowest Point) of the Parabola if the Parabola opens to the Right or Upwards. The Maximum Point (Highest Point) of the Parabola if the Parabola opens to the Left or Downwards.
  • Focus (F) - The focus is a unit leftward or rightward, below or above the vertex. A point inside of the vertex.
  • Directrix (d) - The directrix is a unit leftward or rightward, below or above the vertex opposite of the Focus.
  • Axis of Symmetry - This line divides the parabola into two parts which are mirror images of each other.
  • Latus Rectum - The latus rectum of a parabola is the chord that is passing through the focus of the parabola and is perpendicular to the axis of symmetry of the parabola. The Length of the Latus Rectum is LL' = 4a
• Standard equations of parabola with vertex at (0,0)

  • y^2 = 4ax (opens to the right)
  • y^2 = -4ax (opens to the left)
  • x^2 = 4ay (opens upwards)
  • x^2 = -4ay (opens downwards)
• Standard equations of parabola with vertex at (h,k)


  • (y - k)^2 = 4a(x - h) (opens to the right)
  • (y - k)^2 = -4a(x - h) (opens to the left)
  • (x - h)^2 = 4a(y - k) (opens upwards)
  • (x - h)^2 = -4a(y - k) (opens downwards)
• Ellipse


  The set of all points in a plane, the sum of whose distances from two fixed points, called foci, is a constant.
• Circle as a special case of an ellipse


  A circle is a special case of an ellipse in which a = b
• Parts of an ellipse

  • Major Axis (a) - The major axis is the longest diameter of the ellipse, going through the center from one end to the other, at the broad part of the ellipse. Half of the major axis is called semi-major axis.
  • Minor Axis (b) - the minor axis is the shortest diameter of the ellipse, crossing through the center at the narrowest part of the ellipse, Half of the minor axis is called semi-minor axis.
  • Center (C) - This is the intersection of the two axes of symmetry. This is the midpoint of the foci, and also the midpoint between the vertices and co-vertices.
  • Vertices (V) - This is the end points of the major axis.
  • Co-Vertices (B) - This is the end points of the minor axis.
  • Foci (F) - The foci are always inside the ellipse and are contained by the major axis.
  • Latera Recta (LR) - A chord passing through a focus of an ellipse which is perpendicular to its major axis.
• Formulas for ellipse parts


  a - the distance between the Vertices and the Center.<|>b - the distance between the Co-Vertices and the Center.<|>c - the distance between the Foci and the Center.<|>b^2/a - the distance between the Latera Recta and the Center.
• Standard equation of ellipse with center at (0,0)

  x^2/a^2 + y^2/b^2 = 1
• Major Axis


  The longest diameter of the ellipse, crossing through the center at the broad part of the ellipse. Half of the major axis is called semi-major axis.
• Minor Axis


  The minor axis is the shortest diameter of the ellipse, crossing through the center at the narrowest part of the ellipse, Half of the minor axis is called semi-minor axis.
• Center


  This is the intersection of the two axes of symmetry. This is the midpoint of the foci, and also the midpoint between the vertices and co-vertices.
• Vertices


  This is the end points of the major axis.
• Co-Vertices


  This is the end points of the minor axis.
• Foci


  The foci are always inside the ellipse and are contained by the major axis.
• Latera Recta


  A chord passing through a focus of an ellipse which is perpendicular to its major axis.
• a


  The distance between the Vertices and the Center.
• b


  The distance between the Co-Vertices and the Center.
• c


  The distance between the Foci and the Center.
• b^2/a


  The distance between the Latera Recta and the Center.
• Standard Equation of Ellipse with C(0,0) and Formulas

  • Equation
  • Major Axis
  • Minor Axis
  • Center
  • Vertices
  • Co-Vertices
  • Foci
  • Endpoints of Latera Recta
  • Length of the Latera Recta
  • Orientation
• Standard Equation of Ellipse with C(h,k) and Formulas

  • Equation
  • Center
  • Vertices
  • Co-Vertices
  • Foci
  • Endpoints of Latera Recta
  • Length of the Major Axis
  • Length of the Minor Axis
  • Length of the Latera Recta
  • Orientation
• Equation of Hyperbola


  • Standard Equation (Horizontal Transverse Axis)
  • Standard Equation (Vertical Transverse Axis)
  • Center at C(0,0)
  • Center at C(h,k)
• Transforming Standard Equation into General Equation

  1. The General Equation of the Conic Section is Ax^2 + By^2 + Cx + Dy + E = 0

  2. Example of General Equation of Conic Section