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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
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.
Angle of slice picks the curve, distance rule defines it, and the equation pattern names it.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.