PRE CALCULUS

• Inventor of Conic Sections:

  1. Apollonius of Perga
  2. Menaechmus
• Conic sections are obtained from the intersection between a double-napped cone and a plane
• Conic Sections

  A) parabola

  B) circle

  C) ellipse

  D) hyperbola

  
• Parabolas are formed when the plane is parallel to the generating line one cone
• Ellipses are formed when the plane intersects the one cone at an angle other than 90 degrees
• Hyperbolas are formed when the plane is parallel to the axis of revolution or the y-axis
• Circles are formed when the intersection of the plane is perpendicular to the axis of revolution
• Degenerate conic sections are formed when a plane intersects the cone in such a way that is passes through the apex
• John Wallis, English mathematician, was one of the first to describe that all conics can be written in the form: Ax² + Bxy + Cy² + Dx + Ey + F = 0

  The type of the curve formed by this second -degree equation is determined by the discriminant B²-4AC
• Assuming that a conic is non-degenerate, the following conditions hold:

  1. If B²-4AC < 0, the conics is an ellipse
  2. If B²-4AC < 0, B = 0 and A = C, the conics is a circle
  3. If B²-4AC = 0, the conic is a parabola
  4. If B²-4AC > 0, the conic is a hyperbola
• Eccentricity

  • to identify what type conic the equation is
  • characterizes the shape of a conic section
  • it quantifies how "elongated" or "flattened" a conic section is

  Ax² + Bxy + Cy² + Dx + Ey + F = 0

  A) 0

  B) 1

  C) 1

  D) 0

  E) 1

  
• Circle is a set of all coplanar points such that the distance form a fixed point is constant. The fixed point is called the center of the circle.
• Distance Formula

  A) +

  B) -

  
• Distance Formula to Center-Radius form (Standard Form) of a circle:

  A) x-h

  B) y-k

  
• Find the equation of the circle with center at (-3, -2) and a radius of 7 units

  A) h

  B) k

  
• Midpoint Formula:

  A) 2

  B) 2

  C) +

  
• Find center and radius of the standard form and a general form of circle:

  A) - D/2

  B) - E/2

  C) -F

  
• If r² = 0, then the graph is a single point (not a circle)
• If r² < 0, then there is no graph since r is imaginary
• Circle, Point circle, no graph

  A) circle

  B) point circle

  C) no graph

  
• An ellipse is a set of points in a plane whose sum of distances from two fixed points F₂ and F₂ is constant

  
• The sum of the distances of a point on an ellipse to each foci is equal to any point on an ellipse

  
• principal axis is the line passing through the foci (plural of focus) of an ellipse
• vertices are two points on the ellipse that lie on the principal axis
• major axis is the line segment joining the vertices
• center is the midpoint of the two vertices
• minor axis is the line segment that passes through the center and is perpendicular to the major axis
• focal distance is the distance from the center to a focus
• a is the distance from the center to a vertex
• b is the distance from the center to a co-vertex
• c is the focal distance
• Major Axis:

  A) Horizontal

  B) Vertical

  C) a

  D) b

  E) c

  
• Standard from of the Equation of an Ellipse with center at (h, k)

  A) a

  B) b

  C) +

  D) b

  E) a

  F) 1

  
• Whispering Chambers make use of the ellipse's reflective property so that a whisper from one part of the chamber can be heard to the other part of the chamber
• Parabola is the set of all points that are equidistant from a fixed line called the directrix and a fixed point not on the line called focus (plural, foci)

  A) Focus

  B) Directrix

  
• Vertex

  • It is the highest or lowest point of the parabola
  • It is the maximum or minimum of the parabola
  • It is the point of intersection of the parabola and its axis of symmetry
  • It is the coordinates of the vertex are given as (h, k)
• Focus

  • It is the fixed point on the interior of a parabola
  • The distance from the focus to the vertex is given as c
  • The coordinated of the focus are given:

  Upward / Downward : (h, k ± c)

  Left / Right : (h ± c, k)
• Directrix

  • It is the fixed line on the exterior of a parabola
  • it is the distance from the directrix to the vertex is given as c
  • The coordinated of the directrix:

  Upward / Downward : y = k ∓ c

  Left / Right : x = h ∓ c
• Axis of Symmetry

  • It is the line which divides the parabola into two equal halves and passes through both the vertex and the focus

  Upward / Downward: x = h

  Left / Right : y = k
• Latus Rectum

  • It is the line segment connecting two points on the parabola and passes through the focus
  • It is also a line segment perpendicular to the axis

  Length of the LR (latus rectum) : 4c