Constant Sum Magic

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Why does the ellipse behave this way?

Visual intuition: an ellipse is the set of points P whose distances to two foci always add to the same value.

Move P along the curve; rulers show \(PF_1\) and \(PF_2\). Their sum stays constant—this is the constant-sum rule.

Key Points:

  • F₁ and F₂ are called the foci.
  • For every P on the ellipse, \(PF_1 + PF_2\) is constant.
  • String-and-pins demo: string length = major axis (e.g., 12 cm when pins are 8 cm apart).

Formal Definition

Ellipse

An ellipse is the set of all points in a plane for which the sum of the distances from two fixed points, the foci, is constant.

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Axes & Parts

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Terminology

An ellipse has two perpendicular axes that meet at the centre.

Key Points:

  • Centre: intersection point of the two axes.
  • Major axis: longest chord through the centre.
  • Minor axis: shortest chord through the centre.
  • Vertices: endpoints of each axis.
  • Foci: two internal points on the major axis.
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Key Length Relation

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

Right-angle geometry inside the ellipse gives this Pythagorean-like link.

Variable Definitions

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

Applications

Deriving standard equation

Replace \(c^{2}\) with \(a^{2}-b^{2}\) in the distance rule to obtain the canonical form.

Calculating eccentricity

Use \(e=\frac{c}{a}\) once \(c\) is found from the relation.

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Standard Form

Horizontally oriented ellipse diagram

Ellipse centred at origin

The canonical equation is \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \).

Centre \( (0,0) \); every point \( (x,y) \) obeying it lies on the curve.

Key Points:

  • Called the standard form of an ellipse.
  • Major axis length \(2a\) lies along the x-axis.
  • Minor axis length \(2b\) lies along the y-axis.
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Deriving x²/a² + y²/b² = 1

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

Start with the constant focal-distance sum.

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

Insert distances from \((-c,0)\) and \((c,0)\).

3
\[2a\sqrt{(x-c)^{2}+y^{2}}=4a^{2}-2xc\]

Rearrange, isolate one radical, then square once.

4
\[a^{2}y^{2}=b^{2}(a^{2}-x^{2})\]

Square again, simplify, use \(a^{2}=b^{2}+c^{2}\).

5
\[\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\]

Divide throughout; obtain the standard form of an ellipse.

Key Insight:

Successive squaring removes radicals; substituting \(a^{2}=b^{2}+c^{2}\) converts focus-based terms into the axis lengths \(a\) and \(b\).

Label the Ellipse

Drag each term to its correct location to strengthen your ellipse vocabulary.

Draggable Items

Focus
Major Axis
Minor Axis
Centre
Vertex

Drop Zones

F1 spot

Long horizontal line

Short vertical line

Origin

Endpoint of major axis

Tip:

Need a hint? Recall which part is the longest.

Multiple Choice Question

Question

When the eccentricity \(e = \frac{c}{a}\) tends to 0, the ellipse approaches which shape?

1
A perfect circle
2
A hyperbola
3
A line segment of zero length
4
Its minor axis equals c but shape unchanged

Hint:

Remember: both foci merge at the centre when \(c = 0\).

Key Takeaways

Definition: Sum of distances to two foci stays constant.

Parts: major & minor axes, vertices, centre, foci.

Key relation: \(a^{2}=b^{2}+c^{2}\) links axes to focal length.

Standard form: \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) (horizontal) and its vertical twin.

Eccentricity \(e=\frac{c}{a}\) measures oval-ness, \(0\le e<1\).

Next Steps

Try plotting real-world orbits and measuring their eccentricities!

Thank You!

We hope you found this lesson informative and engaging.