The Ellipse When circles reach for the horizon.

All Points with Equal Sum

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Ellipse: points whose distances to two foci add to a constant.

Geometric Definition of an Ellipse

Place two fixed points called foci, \(F_1\) and \(F_2\).

Any point \(P\) where \(PF_1 + PF_2\) stays constant lies on the ellipse.

Thus, an ellipse is the locus defined by this constant-sum rule.

Key Points:

  • Constant value must be greater than the distance \(F_1F_2\).
  • Closer foci make the ellipse rounder; farther foci stretch it.
  • Quiz: Which everyday running track follows this two-focus rule?

Name the Parts

Ellipse diagram

Locate each feature on an ellipse

Find the centre first, then use it to spot every other part.

Key Points:

  • Major axis: longest chord through the centre.
  • Minor axis: shorter chord perpendicular to the major axis.
  • Vertices: endpoints of each axis.
  • Centre: intersection of the two axes.
  • Foci: two fixed interior points on the major axis.

Meet a, b and c

\[c^{2}=a^{2}-b^{2}\]

In any origin-centred ellipse, the three parameters relate like Pythagoras.

Variable Definitions

a Semi-major axis length
b Semi-minor axis length
c Focus distance from centre

Applications

Consistency Check

Test numbers: does \(4^{2}=5^{2}-3^{2}\)? Yes, so they form an ellipse.

Locate Foci

Compute \(c=\sqrt{a^{2}-b^{2}}\) then plot points \((\pm c,0)\).

Visualising a, b, c

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Ellipse annotated with \(2a, 2b\) and \(c\).

From Symbols to Lengths

An ellipse has three key lengths measured from its centre.

Seeing them on the graph links symbols to real distances.

Key Points:

  • \(2a\): full major-axis length.
  • \(2b\): full minor-axis length.
  • Foci are \(c\) units from centre on the major axis.

Standard Equation

1
\[PF_{1}+PF_{2}=2a\]

Start with the constant-sum definition of an ellipse.

2
\[\sqrt{(x+c)^{2}+y^{2}}+\sqrt{(x-c)^{2}+y^{2}}=2a\]

Express each distance using the distance formula.

3
\[\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\]

Square, simplify, and rearrange to obtain the standard form.

Key Insight:

Any \((x,y)\) that satisfies the equation lies on the ellipse.

Latus Rectum

Latus Rectum Diagram

Definition & Formula

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

Its length is \(2\,\frac{b^{2}}{a}\).

Example: with \(a = 5\) and \(b = 3\), length = \(2\,\frac{3^{2}}{5} = \frac{18}{5}\) units.

Key Points:

  • Chord through a focus
  • Perpendicular to the major axis
  • Length \(2b^{2}/a\)

Key Takeaways

Ellipse essentials

Definition

Every point keeps the sum of distances to two foci constant.

Axes & Centre

Major axis length \(2a\) is longest, minor axis \(2b\) shortest; they intersect at the centre.

Equation

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

Parameter Link

Focus distance obeys \(c^{2}=a^{2}-b^{2}\).

Interactive Insight

Changing \(a\) or \(b\) smoothly stretches or squeezes the curve.