Diving into Ellipses Tracing perfect ovals from chalkboards to cosmic orbits.

What is an Ellipse?

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Fig 10.20 | Ellipse with foci \(F_1,F_2\); blue segments keep constant sum.

Formal Definition (Definition 4)

Definition 4: An ellipse is the locus of points whose distances to two foci add to a constant.

Thus, for every point \(P\), \(PF_1 + PF_2 = 2a\), a fixed length greater than \(F_1F_2\).

Key Points:

  • \(F_1\) and \(F_2\) are the fixed foci.
  • Blue segments in Fig 10.20 mark \(PF_1\) and \(PF_2\).
  • Their sum stays constant (\(2a\)), keeping \(P\) on the ellipse.

Key Parts of an Ellipse

Fig 10.21 Ellipse

Fig 10.21 — Ellipse showing major and minor axes

Centre, Axes & Vertices

Look at Fig 10.21 and match each label. After this, you should name the centre, vertices, major and minor axes with ease.

Key Points:

  • Major axis — longest line through both foci.
  • Minor axis — shortest line, perpendicular to major axis at the centre.
  • Vertices — two endpoints of the major axis.
  • Centre — midpoint of the line joining the foci.

Meet a, b and c

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Parameters of an Ellipse

In Fig 10.22 and Fig 10.23, three numbers fully describe any ellipse drawn with centre O.

Know them, and you can link the semi-axes to the focus distance.

Key Points:

  • \(a\): semi-major axis
  • \(b\): semi-minor axis
  • \(c\): centre \(\rightarrow\) focus distance
  • Relationship: \(a^{2}=b^{2}+c^{2}\)

Proving a² = b² + c²

1
\[PF_1+PF_2 = 2a\]

Place P on the major axis; ellipse definition gives the sum as the full length \(2a\).

2
\[QF_1+QF_2 = 2\sqrt{b^{2}+c^{2}}\]

Pick Q on the minor axis; each focal distance forms a right triangle, giving \(\sqrt{b^{2}+c^{2}}\).

3
\[2a = 2\sqrt{b^{2}+c^{2}}\]

Since the sum is constant for every point, equate results from P and Q.

4
\[a^{2}=b^{2}+c^{2}\]

Square both sides to relate semi-major, semi-minor, and focal lengths.

Key Insight:

Comparing sums at symmetric points links the ellipse’s constant distance property directly to \(a\), \(b\), and \(c\).

Standard Equations

Ellipse diagram (Fig 10.24)

Fig 10.24 • Ellipse with centre at O(0,0)

Centre at (0, 0)

A centred ellipse has two orientations decided by its major axis.

Here \(a\) is the semi-major length and \(b\) the semi-minor, with \(a > b\).

Key Points:

  • Major axis on x-axis: \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \)
  • Major axis on y-axis: \( \frac{x^{2}}{b^{2}} + \frac{y^{2}}{a^{2}} = 1 \)

Deriving x²/a² + y²/b² = 1

Fig 10.25: Ellipse derivation diagram

Fig 10.25 – Point P on an ellipse with foci \(F_1,F_2\)

From Distance Rule to Standard Equation

Fig 10.25 places \(P(x,y)\) on an ellipse whose foci are \(F_1(-c,0)\) and \(F_2(c,0)\).

By definition, the sum of distances satisfies \(PF_1 + PF_2 = 2a\).

Key Steps:

  • Substitute \(PF_1=\sqrt{(x+c)^2+y^2}\) and \(PF_2=\sqrt{(x-c)^2+y^2}\).
  • Square once to remove one root; rearrange terms.
  • Square again to clear remaining radical.
  • Use \(b^{2}=a^{2}-c^{2}\) to obtain \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\).

Quick Properties

Main Points

  1. 1 Symmetric about both x- and y-axes.
  2. 2 Foci lie on the major axis, equidistant from the centre.
  3. 3 Larger denominator ⇒ major axis in \( \dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 \).

Key Highlights

  • Sketch one quadrant; mirror using symmetry.
  • Locate foci at \((\pm c,0)\) or \((0,\pm c)\) along the major axis.
  • Identify the major axis first; other features follow.

Latus Rectum

Latus Rectum diagram

Chord Through the Focus

The latus rectum is the chord that passes through a focus and is perpendicular to the major axis.

Knowing its length lets us quickly compare the “width” of different ellipses at their foci.

Key Points:

  • Length \(= 2\frac{b^{2}}{a}\)
  • Uses semi-major axis \(a\) and semi-minor axis \(b\).
  • Referenced in Sec. 10.5.4 and Fig 10.26.

Worked Example 1

Example 9 — Find foci, vertices, eccentricity and latus rectum of \( \frac{x^{2}}{25}+\frac{y^{2}}{9}=1 \).

1

Identify semi-axes

Compare with \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) to get \(a=5,\,b=3\).

2

Compute focal distance

\(c^{2}=a^{2}-b^{2}=25-9=16 \Rightarrow c=4\).

3

Locate foci & vertices

Major axis lies on \(x\)-axis ⇒ foci at \((\pm4,0)\); vertices at \((\pm5,0)\).

4

Eccentricity & latus rectum

\(e=\frac{c}{a}=\frac{4}{5}=0.8\); length of latus rectum \(=\frac{2b^{2}}{a}=\frac{18}{5}\).

Pro Tip:

For any ellipse centred at the origin, remember \(c^{2}=a^{2}-b^{2}\).

Horizontal vs Vertical

Horizontal Ellipse

\( \dfrac{(x-h)^2}{a^2} + \dfrac{(y-k)^2}{b^2} = 1,\; a>b \)
Major axis \(2a\) runs along the x-axis.
Foci \((h\pm c,\,k)\), \( c=\sqrt{a^2-b^2} \).
Example 10: \( \dfrac{x^{2}}{25} + \dfrac{y^{2}}{9}=1 \).

Vertical Ellipse

\( \dfrac{(x-h)^2}{b^2} + \dfrac{(y-k)^2}{a^2} = 1,\; a>b \)
Major axis \(2a\) runs along the y-axis.
Foci \((h,\,k\pm c)\).
Example 10: \( \dfrac{x^{2}}{9} + \dfrac{y^{2}}{25}=1 \).

Key Similarities

Same centre \((h,k)\).
Relation \(c^2 = a^2 - b^2\) holds.
Eccentricity \(e=\dfrac{c}{a}\) identical.
Latus rectum length \( \dfrac{2b^2}{a} \) same.

Find that Equation

Use vertices & foci to model each ellipse.

1

Example 11 – horizontal axis

Vertices at \((\pm13,0)\) give \(a=13\). Foci at \((\pm5,0)\) give \(c=5\). Then \(b=\sqrt{a^{2}-c^{2}}=\sqrt{169-25}=12\). Equation: \(\dfrac{x^{2}}{169}+\dfrac{y^{2}}{144}=1\).

2

Example 12 – vertical axis

Major axis length \(20\Rightarrow a=10\). Foci distance \(5\Rightarrow c=5\). Hence \(b^{2}=a^{2}-c^{2}=100-25=75\). Equation: \(\dfrac{x^{2}}{75}+\dfrac{y^{2}}{100}=1\).

Pro Tip:

Remember: \(b^{2}=a^{2}-c^{2}\) for every ellipse — your shortcut to quick equations.

Multiple Choice Question

Question

Identify the standard form of an ellipse whose major axis lies on the y-axis.

1
\( \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1 \)
2
\( \frac{x^{2}}{4} + \frac{y^{2}}{25} = 1 \)
3
\( \frac{x^{2}}{16} + \frac{y^{2}}{9} = 1 \)

Hint:

The larger denominator shows the major axis. In Exercise 10.3 intro we learnt: if it is under \(y^{2}\), the axis is vertical.

Ellipse Essentials

An ellipse is the set of points whose distances to two foci sum to a constant.

Semi-major \(a\), semi-minor \(b\), focal length \(c\) satisfy \(c^{2}=a^{2}-b^{2}\).

Standard forms: \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) and \( \frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1 \).

Eccentricity \(e=\frac{c}{a}\); length of latus rectum \( \frac{2b^{2}}{a} \).

Use solved examples, then attempt the practice set to consolidate learning.

Next Steps

Review the summary, finish the practice set, and bring questions to the next class.

Thank You!

We hope you found this lesson informative and engaging.