Pre-Calculus • Lesson 3 • Ellipse

Basic Geometry of an Ellipse

  • Definition
    • An ellipse is the set of all points P(x,y)P(x,y) in the plane whose sum of the distances to two fixed points (the foci) is a constant.
  • Key Parameters
    • aa : semi-major axis (always the larger of the two denominators’ square-roots).
    • bb : semi-minor axis (smaller denominator’s square-root).
    • cc : focal distance, defined by c2=a2−b2c^2 = a^2 - b^2 (note: a > b > 0).
    • Eccentricity e = \dfrac{c}{a}\;(0 < e < 1) measures the “ovalness.”
  • Standard Forms & Elements
    • 1. Horizontal major axis
      (x−h)2a2+(y−k)2b2=1\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1
      • Center : (h,k)(h,k)
      • Vertices : V(h!+!a,k)V(h!+!a,k) and V′(h!−!a,k)V'(h!-!a,k)
      • Foci : F(h!+!c,k)F(h!+!c,k) and F′(h!−!c,k)F'(h!-!c,k)
      • Co-vertices : B(h,k!+!b)B(h,k!+!b) and B′(h,k!−!b)B'(h,k!-!b)
      • Principal (major) axis : line y=ky = k
    • 2. Vertical major axis
      (x−h)2b2+(y−k)2a2=1\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1
      • Center : (h,k)(h,k)
      • Vertices : V(h,k!+!a)V(h,k!+!a) and V′(h,k!−!a)V'(h,k!-!a)
      • Foci : F(h,k!+!c)F(h,k!+!c) and F′(h,k!−!c)F'(h,k!-!c)
      • Co-vertices : B(h!+!b,k)B(h!+!b,k) and B′(h!−!b,k)B'(h!-!b,k)
      • Principal (major) axis : line x=hx = h
  • Length of the Latus Rectum
    • The latus rectum is the focal chord ⟂ to the major axis. Its length is
      LR=2b2aLR = \frac{2b^2}{a} (same formula for either orientation).
  • Endpoints of the Latera Recta
    • Horizontal case: (h!±!c,  k!±!b2a)\bigl(h!\pm!c,\;k!\pm!\dfrac{b^2}{a}\bigr)
    • Vertical case : (h!±!b2a,  k!±!c)\bigl(h!\pm!\dfrac{b^2}{a},\;k!\pm!c\bigr)

Worked Example 1

Equation : (x−5)249+(y−4)225=1\frac{(x-5)^2}{49}+\frac{(y-4)^2}{25}=1 (major axis horizontal)

  • Identify parameters
    a2=49  ⟹  a=7a^2 = 49 \;\Longrightarrow\; a = 7
    b2=25  ⟹  b=5b^2 = 25 \;\Longrightarrow\; b = 5
    c2=a2−b2=49−25=24  ⟹  c≈4.90c^2 = a^2 - b^2 = 49 - 25 = 24 \;\Longrightarrow\; c \approx 4.90
    Center C(5,4)C(5,4)
  • Vertices
    • V(5+7,4)=(12,4)V(5+7,4) = (12,4)
    • V′(5−7,4)=(−2,4)V'(5-7,4) = (-2,4)
  • Foci
    • F(5+4.90,4)≈(9.90,4)F(5+4.90,4) \approx (9.90,4)
    • F′(5−4.90,4)≈(0.10,4)F'(5-4.90,4) \approx (0.10,4)
  • Co-vertices
    • B(5,4+5)=(5,9)B(5,4+5)=(5,9)
    • B′(5,4−5)=(5,−1)B'(5,4-5)=(5,-1)
  • Length of latus rectum
    LR=2b2a=2(25)7≈7.14 unitsLR = \dfrac{2b^2}{a} = \dfrac{2(25)}{7} \approx 7.14\text{ units}
  • Endpoints of the latera recta y-offset=b2a=257=3.57y\text{-offset}=\dfrac{b^2}{a}=\dfrac{25}{7}=3.57
    1. (9.90,  4+3.57)=(9.90,7.57)(9.90,\;4+3.57)=(9.90,7.57)
    2. (9.90,  4−3.57)=(9.90,0.43)(9.90,\;4-3.57)=(9.90,0.43)
    3. (0.10,  7.57)(0.10,\;7.57)
    4. (0.10,  0.43)(0.10,\;0.43)
  • Principal axis : y=4y = 4 (horizontal line through the center)

Worked Example 2

Equation : (x+3)236+y264=1\frac{(x+3)^2}{36}+\frac{y^2}{64}=1 (major axis vertical)

  • Identify parameters
    a2=64  ⟹  a=8a^2 = 64 \;\Longrightarrow\; a = 8
    b2=36  ⟹  b=6b^2 = 36 \;\Longrightarrow\; b = 6
    c2=a2−b2=64−36=28  ⟹  c≈5.29c^2 = a^2 - b^2 = 64 - 36 = 28 \;\Longrightarrow\; c \approx 5.29
    Center C(−3,0)C(-3,0)
  • Vertices
    • V(−3,0+8)=(−3,8)V(-3,0+8)=(-3,8)
    • V′(−3,−8)V'(-3,-8)
  • Foci
    • F(−3,5.29)F(-3,5.29)
    • F′(−3,−5.29)F'(-3,-5.29)
  • Co-vertices
    • B(−3+6,0)=(3,0)B(-3+6,0)=(3,0)
    • B′(−9,0)B'(-9,0)
  • Length of latus rectum
    LR=2b2a=2(36)8=9 unitsLR = \dfrac{2b^2}{a}=\dfrac{2(36)}{8}=9\text{ units}
  • Endpoints of the latera recta x-offset=b2a=368=4.5x\text{-offset}=\dfrac{b^2}{a}=\dfrac{36}{8}=4.5
    1. (−3+4.5,  5.29)=(1.5,5.29)(-3+4.5,\;5.29)=(1.5,5.29)
    2. (−3−4.5,  5.29)=(−7.5,5.29)(-3-4.5,\;5.29)=(-7.5,5.29)
    3. (1.5,−5.29)(1.5,-5.29)
    4. (−7.5,−5.29)(-7.5,-5.29)
  • Principal axis : x=−3x = -3 (vertical line through the center)

Quick Reference Formulas

  • Relationship among parameters: c2=a2−b2c^2=a^2-b^2
  • Eccentricity: e=cae=\dfrac{c}{a}
  • Area: A=πabA=\pi a b
  • Perimeter (approx.): P  ≈  π[3(a+b)−(3a+b)(a+3b)1]P\;\approx\;\pi\bigl[\tfrac{3(a+b)-\sqrt{(3a+b)(a+3b)}}{1}\bigr] (Ramanujan’s first approximation)

Practice Problems ("LET’S TRY!")

  1. 4x2+8y2−4x−24y−13=04x^2+8y^2-4x-24y-13=0
    • Complete the square to convert to standard form and identify all key elements.
  2. Center : origin; focus : (0,5)(0,5); endpoint of minor axis : (7,0)(7,0).
    • Determine a,b,ca,b,c and write the equation.
  3. Center : origin; passes through (8,3)(8,3); length of the major axis =20=20.
    • Hint : major axis length gives 2a2a. Use the point to find bb.

Conceptual & Practical Notes

  • Orientation test : The larger denominator sits under the variable whose axis is the major axis.
    Example : If (x−h)2121+(y−k)225=1\frac{(x-h)^2}{121}+\frac{(y-k)^2}{25}=1, then a=11a=11, major axis is horizontal.
  • Degenerate cases occur when b→0b \to 0, collapsing the ellipse to a line segment of length 2a2a.
  • Applications
    • Planetary orbits (Kepler’s First Law): planets move in elliptical orbits with the Sun at one focus.
    • Whispering galleries & lithotripsy: sound/light rays emanating from one focus reflect to the other.
  • Ethical / philosophical: Modeling real-world phenomena (e.g.
    epidemics, economics) with ellipses reminds us that elegant mathematics can describe, but also oversimplify, complex realities—critical thinking is essential.