Ellipse

From the locus definition to exam-style line intersection techniques—learn the ellipse with diagrams, formulas, an interactive calculator, and a practice quiz.

Advanced

Conic Sections

Calculator + Quiz

Quick Facts

Keep these on a sticky note while solving ellipse problems.

Definition

Distance sum to foci is constant (2a)

Standard form

x²/a² + y²/b² = 1 (a>b>0) or x²/b² + y²/a² = 1

Parameter relation

c² = a² − b², e = c/a (0<e<1)

Area

A = πab

Latus rectum

Length = 2b²/a

Ellipse Calculator (Interactive)

Compute parameters, foci, vertices, area, and latus rectum—plus see a real-time SVG preview.

a (semi-major)

b (semi-minor)

Enter parameters and click Calculate to see results.

Core Knowledge

1) Definition & Standard Equations

From “sum of distances” to the standard form with a, b, c.

Locus definition

An ellipse is the set of points P such that the sum of distances to two fixed points (the foci) is constant:

PF_1 + PF_2 = 2a

Here a is the semi-major axis length. The foci are separated by distance 2c.

Standard equations

Horizontal major axis (foci on x-axis):

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\quad (a>b>0)

Vertical major axis (foci on y-axis):

\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\quad (a>b>0)

Parameter relation

Once you know a and b, the focus parameter is fixed:

c^2 = a^2 - b^2

Ellipse anatomy (a, b, c, foci, axes)

The parameters satisfy c² = a² − b², with foci at (±c, 0) (horizontal major axis).

Latus rectum (through focus ⟂ major axis)

The latus rectum has length 2b²/a. It is the shortest focal chord and appears often in exam problems.

Vertices & axis lengths

For a horizontal ellipse, the vertices are (±a, 0) and the co-vertices are (0, ±b).

Major axis length

Minor axis length

Eccentricity

Eccentricity measures how “stretched” the ellipse is:

e=\frac{c}{a}\quad (0<e<1)

Larger e means the foci are farther apart and the ellipse is flatter.

Eccentricity: larger e ⇒ flatter ellipse

Eccentricity is e = c/a, with 0 < e < 1. When e increases, the foci move farther apart and the ellipse becomes less “round”.

2) Geometric Properties (Exam Toolkit)

Focal radius, latus rectum, and the “shape” knobs (a, b, c, e).

Focal radius (horizontal ellipse)

Let P(x₀, y₀) lie on the ellipse, and foci be F₁(−c,0), F₂(c,0). The two focal radii satisfy:

PF_1 = a + ex_0,\qquad PF_2 = a - ex_0

Memory cue: “left plus, right minus” when foci are on the x-axis.

Latus rectum

The chord through a focus perpendicular to the major axis has constant length:

\text{Latus rectum length }= \frac{2b^2}{a}

It is the shortest focal chord and is frequently used in parameter problems.

Common “shape reasoning”

Fix a: larger e means smaller b and a flatter ellipse.
Fix c: larger a means smaller e and a more circular ellipse.
Always check: a > b > 0, and c² = a² − b².

Line–ellipse intersection (eliminate + discriminant)

Substitute the line into the ellipse to get a quadratic. Use the discriminant to determine whether it is a secant, tangent, or exterior line.

Vertices & foci summary

Horizontal major axis

Vertices: (±a,0)

Foci: (±c,0)

Vertical major axis

Vertices: (0,±a)

Foci: (0,±c)

Tip: Decide the orientation first (where the foci lie), then write down vertices and foci in one line.

3) Line–Ellipse Intersections (Δ + Vieta)

System solving, discriminant classification, and typical invariant tricks.

Intersection workflow

Choose a line form (slope-intercept or point-slope).
Substitute into the ellipse to eliminate y (or x).
You get a quadratic in one variable; use Δ to classify.
Use Vieta for x₁+x₂ and x₁x₂ (or y₁+y₂, y₁y₂).

Discriminant classification

\Delta>0\Rightarrow \text{secant (2 intersections)},\quad \Delta=0\Rightarrow \text{tangent},\quad \Delta<0\Rightarrow \text{no real intersection}.

Vieta inside geometry

Many invariants (fixed midpoint, fixed area, slope product constant) come from expressing everything using x₁+x₂ and x₁x₂. This avoids solving for the two intersection points explicitly.

Typical move

Convert chord length / triangle area / midpoint conditions into algebraic expressions of Vieta sums/products.

Focal chord ideas

A focal chord is any chord passing through a focus. Typical problems ask for:

Chord length in terms of slope/angle.
Area of a focal triangle.
Ratio constraints like PF₁:PF₂ or PF₁±PF₂.

Always start by writing the definition PF₁ + PF₂ = 2a and the relation e=c/a.

Fixed-point / fixed-value templates

Common exam templates often reduce to:

Slope product constant from symmetric intersection pairs.
Area constant from a fixed base or fixed distance to an axis.
Midpoint constraint via chord midpoint formula.

Strategy: express everything in terms of one parameter (slope k or intercept m), then eliminate it using the given condition to force an invariant.

4) Extensions (Optics + Astronomy)

Why ellipses show up in mirrors and planetary motion.

Optical reflection property

An ellipse reflects rays from one focus to the other: if a light ray starts at F₁, reflects off the ellipse, it passes through F₂. This is why “whispering galleries” and certain reflective designs use ellipses.

Exam connection: problems may encode reflection as an angle condition which you translate into equal tangential angles at the point of reflection.

Kepler’s First Law (planetary orbits)

In classical mechanics, bound two-body orbits are ellipses with the central body at one focus. This is Kepler’s first law. While high-school analytic geometry doesn’t derive orbital mechanics, the same focal distance structure appears in many applications.

Useful intuition: e indicates “how non-circular” an orbit is—small e is near-circular, large e is elongated.

A compact summary (for speed)

One-liner

Ellipse: PF₁ + PF₂ = 2a.

Parameters

c² = a² − b², e = c/a.

Lengths

Area πab, latus rectum 2b²/a.

5) Common Pitfalls

Small mistakes that cost big points in conic problems.

1) Forgetting orientation

Before writing foci and vertices, decide whether the major axis is horizontal or vertical. Many “wrong sign / swapped denominator” errors come from this.

2) Using the wrong relation for c

Ellipse: c² = a² − b². (Hyperbola uses c² = a² + b².)

3) Solving intersections too early

If a question asks for an invariant, don’t solve for both intersection points explicitly. Use Vieta (sum/product) first.

Practice Quiz

Questions

Correct

Accuracy

An ellipse is the set of points whose distances to two fixed points have a constant sum. What is this constant commonly written as?

Not attempted

For an ellipse with semi-axes a = 5 and b = 3 (a > b), what is c?

Not attempted

Which range is correct for ellipse eccentricity e?

Not attempted

For a horizontal ellipse, the foci are located at:

Not attempted

The area of an ellipse with semi-axes a and b equals:

Not attempted

For x²/a² + y²/b² = 1 (a > b > 0), which statement is true?

Not attempted

The latus rectum length of an ellipse (horizontal or vertical) is:

Not attempted

A line intersects an ellipse in 2, 1, or 0 real points depending on:

Not attempted

If e increases (still less than 1), the ellipse becomes:

Not attempted

Which statement about ellipse reflection is correct?

Not attempted