What it shows
The four conic sections (circle, ellipse, parabola and hyperbola) are the curves you get when a flat plane cuts a double cone. They can also be described by one rule: a point P moves so that its distance to a fixed point, the focus F, is always e times its distance to a fixed line, the directrix. The number e is the eccentricity: e = 0 for a circle, between 0 and 1 for an ellipse, exactly 1 for a parabola, and more than 1 for a hyperbola.
This page draws each curve from its standard equation and marks the foci (red dots), directrices (dashed orange lines) and latus rectum (the green chord through a focus, square to the axis). A point P moves along the curve so you can check the defining distances yourself.
It follows conic sections in Class 11 maths. The unit circle page covers circles centred at the origin and their angles; the argand plane page uses the same plane for complex numbers; and the graphing calculator can draw any curve you type, conics included.
How to use it
- Pick a conic: Circle, Ellipse, Parabola, Hyperbola, or Slice a cone.
- Slide the numbers. Circle: centre (h, k) and radius r. Ellipse and hyperbola: a and b (for an ellipse, whichever is bigger lies along the major axis). Parabola: the focus distance a and which way it opens. A hyperbola can open left–right or up–down.
- Move P along the curve with the bottom slider. The line under the picture gives the distances: PF₁ + PF₂ for an ellipse, PF₁ − PF₂ for a hyperbola, PF and PD for a parabola.
- Slice a cone: set the cone’s half-angle α and tilt the plane to angle β from the axis. The lower picture shows the curve that is cut.
Formulas used
- Ellipse x²/a² + y²/b² = 1 with a > b: c² = a² − b², foci (±c, 0), e = c/a, directrices x = ±a/e, latus rectum 2b²/a.
- Hyperbola x²/a² − y²/b² = 1: c² = a² + b², foci (±c, 0), e = c/a, directrices x = ±a/e, latus rectum 2b²/a, asymptotes y = ±(b/a)x (dashed grey).
- Parabola y² = 4ax: focus (a, 0), directrix x = −a, latus rectum 4a, e = 1.
- Circle (x − h)² + (y − k)² = r²: centre (h, k), radius r.
- Cone slice: β = 90° gives a circle, β > α an ellipse, β = α a parabola and β < α a hyperbola, with e = cos β / cos α.
Worked example
For x²/25 + y²/9 = 1, a = 5 and b = 3, so c = √(25 − 9) = 4. The foci are (±4, 0), e = 4/5 = 0.8, the directrices are x = ±5/0.8 = ±6.25 and the latus rectum is 2 × 9/5 = 3.6. Wherever P is, PF₁ + PF₂ = 10 = 2a.
For x²/9 − y²/16 = 1, c = √(9 + 16) = 5, e = 5/3, the directrices are x = ±1.8 and the latus rectum is 2 × 16/3 ≈ 10.667.
Check yourself
Five questions with fresh numbers: an ellipse’s eccentricity, a parabola’s focus, a hyperbola’s latus rectum, a circle’s radius from its general equation, and which conic a plane parallel to the cone’s side cuts. Nothing is saved.
FAQ
Why is a circle an ellipse? When a = b, c = 0: both foci sit at the centre and e = 0.
Why are there two branches of a hyperbola? The plane is steep enough to cut both halves of the double cone.
Is this from my textbook? It covers the standard school topic with its own examples and pictures. It is an independent study aid.
Where the numbers come from
- Standard coordinate geometry: the standard equations of the circle, ellipse, parabola and hyperbola, their foci, directrices, eccentricity and latus rectum, and the cone section e = cos β / cos α. Written for fibub; no textbook text, figures or exercises are used.
- Checked with NumPy: points sampled from each equation satisfy the focus–directrix property PF = e·PD and the two-focus sum or difference 2a, and the cone-slice eccentricity agrees with a conic fitted to points cut from a double cone.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.