Core Idea

Ellipse

All points \(P\) satisfying \(PF_{1}+PF_{2}=k\), with fixed foci \(F_{1},F_{2}\) and constant \(k > F_{1}F_{2}\).

Key Characteristics:

  • Two special points called foci guide the curve.
  • Sum of distances to the foci is constant for every point.
  • This constant exceeds the focal separation.

Example:

If the sum equalled the focal distance, what locus would form?

Building the Equation

Follow the algebraic trail from definition to equation.

1

Set the Coordinates

Place foci at \(F_1(-c,0)\) and \(F_2(c,0)\). Pick point \(P(x,y)\). Centre at origin, major axis on \(x\)-axis.

2

Apply Distance Formula

Compute \(PF_1=\sqrt{(x+c)^2+y^2}\) and \(PF_2=\sqrt{(x-c)^2+y^2}\). By definition, \(PF_1+PF_2=2a\).

3

Isolate \(x\) and \(y\)

Square twice and simplify to get \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) where \(b^2 = a^2 - c^2\).

Pro Tip:

Remember \(a \gt b\). If \(c=0\), the ellipse collapses into a circle.

Crunching Numbers

1
\[(x+c)^2 + y^2 + (x-c)^2 + y^2 = (2a)^2\]

Square each focus distance and add; constant \(2a\) fixes ellipse size.

2
\[2x^2 + 2y^2 = 4a^2 - 2c^2\]

Expand brackets, then isolate \(x\) and \(y\) terms; constants shift right.

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

Divide by \(a^2\) and set \(b\); standard ellipse form emerges.

Key Insight:

Each algebra step mirrors geometry: squaring fixes focus distances, isolation separates variables, \(a\) and \(b\) reveal semi-axes.

Axis Relations

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

Relation linking semi-major \(a\), semi-minor \(b\), and focal distance \(c\).

Variable Definitions

a Semi-major axis (longest radius)
b Semi-minor axis (shorter radius)
c Distance from centre \(O\) to each focus

Applications

Find Foci

Compute \(c=\sqrt{a^{2}-b^{2}}\) when axes lengths are known.

Check Axis Data

Verify if three given lengths can form an ellipse by testing \(a^{2}=b^{2}+c^{2}\).

Distance Proofs

Supports proof that any point \(P\) satisfies \(PF_{1}+PF_{2}=2a\).

Eccentricity e

\(e = \frac{c}{a}\)

Eccentricity \(e\) quantifies ellipse flatness: \(0 < e < 1\). \(e\) near 0 ⇒ almost circle; \(e\) near 1 ⇒ very elongated.

Earth’s orbit has \(e \approx 0.017\) — practically circular.

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Latus Rectum

Diagram

What is the Latus Rectum?

The latus rectum is a special chord of an ellipse.

It passes through a focus and is perpendicular to the major axis.

Key Points:

  • Definition: chord through a focus \(F(c,0)\) ⟂ major axis.
  • Find its ends by substituting \(x=c\) into \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \).
  • Length \( \text{LR}=2|y|=\frac{2b^{2}}{a} \).
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Elliptical Takeaways

Key facts in one glance

Definition

Set of all points whose distances to two foci add to \(2a\).

Standard Equation

\(\displaystyle \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,\; a>b>0\).

Focus Relation

Semi-axes satisfy \(a^{2}=b^{2}+c^{2}\).

Eccentricity

\(e=\frac{c}{a},\; 0<e<1\) gauges “ovalness”.

Latus Rectum

Through a focus, length \( \frac{2b^{2}}{a}\).