---
title: "Conic Sections: Ellipse, Parabola and Hyperbola"
description: "Meet the three curves a plane cuts from a cone, in both their locus definitions and their standard equations, and read foci and axes off the equation."
canonical: https://lightmysky.com/learn/mathematics/conic-sections-ellipse-parabola-and-hyperbola-mt_V4YSI3a_Eu
source: https://lightmysky.com/learn/mathematics/conic-sections-ellipse-parabola-and-hyperbola-mt_V4YSI3a_Eu.md
retrieved: 2026-09-12
---

> **Agent view.** This is the Markdown twin of the page, for tools and assistants.
> When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md
> API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md
> Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt
> Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json
> Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`.

# Conic Sections: Ellipse, Parabola and Hyperbola

Meet the three curves a plane cuts from a cone, in both their locus definitions and their standard equations, and read foci and axes off the equation.

Subject: Mathematics · Area: Geometry · Ages 17 to 18
Page: https://lightmysky.com/learn/mathematics/conic-sections-ellipse-parabola-and-hyperbola-mt_V4YSI3a_Eu

## Ready when they can

- Match each standard equation to its curve and identify centre, axes and foci
- State the distance property that defines an ellipse, a parabola and a hyperbola
- Recognise a conic from a shifted equation by completing the square in both variables

## Lesson: Slicing a cone into three curves

Every conic is a slice of a cone, and the slicing angle decides the curve. A gentle tilt gives an ellipse, a slice parallel to the side gives a parabola, and a steep cut through both halves gives a hyperbola. Equations show the same split: plus between the squares means ellipse, minus means hyperbola, and a single squared variable means parabola.

Each curve also owns a distance rule. An ellipse fixes the sum of the distances to two foci. A parabola matches one focus to a line called the directrix, point for point. A hyperbola fixes the difference of the distances to two foci.

**Example.** Read x squared over 25 plus y squared over 9 equals 1. Both signs are positive with unlike denominators, so it is an ellipse with a equal to 5 along x, and its vertices are (5, 0) and (minus 5, 0). For the hyperbola x squared over 16 minus y squared over 9 equals 1, c squared equals 16 plus 9, so c equals 5. For the parabola y squared equals 12x, 4p equals 12, so the focus sits 3 units from the vertex.

**Tip.** If squares come expanded with extra x or y terms, complete the square in each variable to reveal the shifted centre. The ellipse (x minus 1) squared over 4 plus (y plus 2) squared over 9 equals 1 centres at (1, minus 2), since x minus 1 and y plus 2 set the shifts.

**Recap.** Angle of slice picks the curve, distance rule defines it, and the equation pattern names it.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Parametric Equations of Curves](https://lightmysky.com/learn/mathematics/parametric-equations-of-curves-mt_7NC7SCeU2P)
- [Completing the Square](https://lightmysky.com/learn/mathematics/completing-the-square-mt_rHmCfzB4Fm)
- [The Equation of a Circle](https://lightmysky.com/learn/mathematics/the-equation-of-a-circle-mt_YARqmi_ioe)

## Opens up

- [Polar Coordinates and Polar Curves](https://lightmysky.com/learn/mathematics/polar-coordinates-and-polar-curves-mt_R5QMUJvA1v)