Further Coordinate Systems

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WFM03 · Chapter 2

Further Coordinate Systems

Ellipses, hyperbolas and parabolas are all the same family of curve viewed from different angles — and once you can write them parametrically, you can find tangents, normals and loci using the same chain-rule trick every single time.

Summary

Ellipse: Cartesian and parametric forms, foci, directrices, eccentricity < 1, and the "string method" definition PS + PS′ = 2a.
Hyperbola: the "twin" of the ellipse with eccentricity > 1, asymptotes, and the same focus-directrix definition PS = ePN.
Parabola: the special case where e = 1 — one focus, one directrix, no centre of symmetry.
Parametric differentiation: finding dy/dx from x(θ) and y(θ) using the chain rule, without ever eliminating the parameter.
Tangents and normals: using the parametric gradient to write down the equation of a tangent or normal at a general point.
Finding a locus: tracking a point that depends on a parameter, then eliminating the parameter to get a Cartesian equation for the path it traces.

The Ellipse

Think of an ellipse as a circle that's been squashed. Instead of one radius, it has two "half-widths" — a along the x-axis and b along the y-axis.

Cartesian & Parametric Forms

Cartesian: x²/a² + y²/b² = 1 | Parametric: x = a cos θ, y = b sin θ

Every point on the ellipse can be written as (a cos θ, b sin θ) for some angle θ — this is just because cos²θ + sin²θ = 1 automatically satisfies the Cartesian equation.

Foci, directrices and eccentricity

An ellipse has two foci (special interior points) at S(ae, 0) and S′(−ae, 0), and two directrices (fixed vertical lines) at x = ±a/e. The eccentricity e tells you how "squashed" the ellipse is — for an ellipse, always e < 1. The three quantities a, b and e are linked by:

Key relationship

b² = a²(1 − e²)

The focus–directrix definition

Here's the deep idea underneath all of this: an ellipse is the set of all points P such that the distance to a focus is a fixed fraction of the distance to the corresponding directrix:

Definition

PS = e · PN

where S is a focus, N is the foot of the perpendicular from P to the corresponding directrix, and this holds true for either focus with its matching directrix.

🧵 Why the "string method" works If you pin a loop of string around both foci and trace it taut with a pencil, you draw a perfect ellipse. Here's the proof buried in the notes: since PS = ePN and PS′ = ePN′, adding these gives PS + PS′ = e(PN + PN′) = e·NN′ = e·(2a/e) = 2a. So the sum of distances from any point on the ellipse to the two foci is always 2a — a constant. That constant string length is exactly 2a.

Worked Example

Q: The ellipse x²/25 + y²/16 = 1 has eccentricity e. Find e, and the coordinates of the foci.

Step 1: Here a² = 25, b² = 16, so a = 5, b = 4.

Step 2: Use b² = a²(1 − e²): 16 = 25(1 − e²) ⇒ 1 − e² = 16/25 ⇒ e² = 9/25 ⇒ e = 3/5.

Step 3: Foci are at (±ae, 0) = (±5 × 3/5, 0) = (±3, 0).

Practice Question

An ellipse has parametric equations x = 6 cos θ, y = 2 sin θ. Find its eccentricity and the equations of its directrices.

The Hyperbola

two open branches

Cartesian & Parametric Forms

Cartesian: x²/a² − y²/b² = 1 | Parametric: x = a cosh θ, y = b sinh θ

Asymptotes

The Parabola

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Also in the full note

Parametric Differentiation
Tangents and Normals
Finding a Locus
What to Memorise
Concepts Checklist
Exam Tips
Foci, directrices, eccentricity

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Further Coordinate Systems