Meet the Ellipse Where stretched circles plot the path of planets.

Formal Definition

Ellipse

An ellipse is all points in a plane whose distances to two fixed foci add to the same constant.

Quick check: If the constant sum is 10 cm and one focus is 6 cm from point P, how far is P from the other focus? Answer: 4 cm

Constant-Sum Property

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Red segments: \(PF_1 + PF_2\) stays constant

One Simple Rule Shapes the Ellipse

An ellipse is the set of all points \(P\) for which the sum of distances to two fixed points—the foci \(F_1\) and \(F_2\)—is constant.

Watch point \(P\) move. As \(PF_1\) shortens and \(PF_2\) lengthens, their combined length never changes, so the path naturally traces the curve.

Key Points:

  • Two fixed points are the foci \(F_1, F_2\).
  • For any point \(P\), \(PF_1 + PF_2 = 2a\) (a constant).
  • Fixing this sum draws the complete ellipse.

Key Parts

https://asset.sparkl.ac/pb/sparkl-vector-images/img_ncert/z0qcfb4l1Eqiiw3lFdySzoCTdHsHrvjjPLRLwqZI.png

Ellipse with labelled axes and vertices

Parts of an Ellipse

Two perpendicular axes cross at centre \(O\), defining the key components of the ellipse.

Key Points:

  • Major axis: longest diameter; runs through widest points.
  • Minor axis: shortest diameter; perpendicular to major axis.
  • Vertices: four points where axes meet the curve.
  • Centre \(O\): midpoint where axes cross; symmetry point.

Semi-Lengths & Focus Gap

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Relating \(a\), \(b\) and \(c\)

An ellipse is governed by three parameters.

Semi-major \(a\) and semi-minor \(b\) are half the major and minor axes.

Focal distance \(c\) measures the centre-to-focus gap.

Key Points:

  • \(a \ge b > 0\)
  • Relation: \(c = \sqrt{a^{2}-b^{2}}\)
  • Flatter ellipse when \(b/a\) is smaller

Why \(a^{2}=b^{2}+c^{2}\)?

Goal: derive the relation linking semi-axes \(a, b\) and focal distance \(c\).

1

Distance formula

Write \(PF_{1}=\sqrt{(x+c)^{2}+y^{2}}\) and \(PF_{2}=\sqrt{(x-c)^{2}+y^{2}}\).

2

Ellipse definition

Set \(PF_{1}+PF_{2}=2a\).

3

Square & simplify

Isolate a radical, square twice, then collect like terms.

4

Relation revealed

\(x\) and \(y\) cancel, leaving \(a^{2}=b^{2}+c^{2}\).

Pro Tip:

Imagine a right triangle with legs \(b\) and \(c\); its hypotenuse is \(a\).

Standard Equations

Horizontal and Vertical Ellipse Forms

Horizontal vs Vertical major axis

Which axis gets \(a^{2}\)?

Both forms satisfy \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\) with \(a \ge b\).

If \(a^{2}\) sits under \(x^{2}\), the major axis is horizontal; if under \(y^{2}\), it is vertical.

Key Points:

  • Horizontal major axis: \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\)
  • Vertical major axis: \( \frac{x^{2}}{b^{2}} + \frac{y^{2}}{a^{2}} = 1\)

Multiple Choice Question

Question

For the ellipse \( \dfrac{x^{2}}{9} + \dfrac{y^{2}}{16} = 1 \), what is the distance between its two foci?

1
7 units
2
\(2\sqrt7\) units
3
8 units
4
4 units

Hint:

First find \(c\) using \(c^{2}=a^{2}-b^{2}\); the focal distance is \(2c\).

Ellipse in a Nutshell

Set of points whose distances to two foci add to a constant.

Standard form: \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \) with \( a > b > 0 \).

Major axis \(2a\), minor axis \(2b\); foci satisfy \( c^{2} = a^{2} - b^{2} \).

Eccentricity \( e = \frac{c}{a} \); always between 0 and 1.

Reflection property: a line from one focus reflects to the other.

Area formula: \( \text{Area} = \pi a b \).

Next Steps

Attempt exercise 10.2 and sketch ellipses with varying \(e\) to solidify concepts.

Thank You!

We hope you found this lesson informative and engaging.