Conic Parameters Explorer

Adjust center, shape parameters, and orientation to see equation geometry update live.

Worked example — Conic Parameters Explorer

ellipse standard form	1
Center	(0,0)
semi-major	4
semi-minor	3
focus distance c	2.645751

These are the values the calculator shows when the page opens. Change any input above to recompute them.

Loading interactive simulation...

Only the parabola gets a directrix drawn, and the vertex sits exactly halfway 🖖

Switch between the presets and one row appears for the parabola alone: the directrix. That is not a gap in the other two — it is the definition. Take parabolaRight: the vertex is at (−2, 0), the focus at (1, 0) and the directrix at x = −5, which puts the vertex exactly 3 from each. That distance is p, and it is why moving p slides focus and directrix apart symmetrically. A parabola is the curve whose points are equally far from a focus and a line, so the ratio is pinned at 1; ellipses and hyperbolas instead report foci offset by c.

Where the name comes from 🖖

The word "conic" is literal: slice a double cone with a flat plane and the edge you cut out is one of these curves. Hold the plane level and you get a circle; tilt it gently for an ellipse; tilt until it runs parallel to the cone's side for a parabola; tilt further so it cuts both halves and you get a hyperbola's two branches. Every shape in this tool is just a different cutting angle.

The focus hidden inside the cone 🖖

A conic drawn as a flat equation has a focus, and a conic sliced from a cone has a cutting plane — these seem unrelated. In 1822 Germinal Dandelin showed they are the same point: fit a sphere snugly inside the cone so it touches both the cone and the cutting plane, and the sphere's contact point is exactly the focus. For an ellipse two such spheres fit, pinning down both of its foci.

Problem solved in full

An ellipse with semi-axes 4 and 3 and foci at c = 2.6458 5 steps
An ellipse with semi-axes 4 and 3 has its foci at c = 2.6458. Derive that, and then the property that makes it an ellipse in the first place.
1. The three lengths are not independent. Stand at the end of the minor axis and the two focal distances are equal, each spanning a right triangle with legs b and c and hypotenuse a — so a² = b² + c², and the Pythagorean relation lives inside every ellipse.
  
  $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a = 4, b = 3$
2. Solving for c gives √7. The tool prints 2.645751, which is that surd.
  
  $a^{2} = b^{2} + c^{2} ⟹ c = 16 - 9 = 7 = 2.645751$
3. Eccentricity is the ratio of c to a, and it is the shape parameter — the one number that says how far from circular the curve is, independent of size.
  
  $e = \frac{c}{a} = \frac{2.645751}{4} = 0.6614$
4. The defining property follows: the sum of the distances to the two foci is the same for every point, and equals 2a. Check it at the end of the major axis, where the distances are a − c and a + c.
  
  $r_{1} + r_{2} = 2 a = 8 for every point on the curve$
5. Eccentricity also classifies the whole family. Zero is a circle, approaching 1 is a parabola, above 1 is a hyperbola — the same equation throughout.
  
  $e \to 0 \Rightarrow circle; e \to 1 \Rightarrow parabola; e > 1 \Rightarrow hyperbola$
Answer

The tool prints a = 4, b = 3 and c = 2.645751. The constant-sum property is the one to hold on to, because it is both the definition and the construction: two pins and a loop of string of length 2a + 2c draw exactly this curve, and that is not a craft trick but the definition made physical. It is also why orbits are ellipses with the sun at a focus and not at the centre — the focus is where the physics is, and the centre is just the midpoint of a drawing.

$c = a^{2} - b^{2} = 7, e = 0.661; r_{1} + r_{2} = 2 a defines the ellipse$

References (1)

Insight block 3 — the sphere that turns a cutting plane into a focus: G. P. Dandelin, "Mémoire sur quelques propriétés remarquables de la focale parabolique." Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Bruxelles 2, 1822.

By the 42Math team · Updated 2026-05-21

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

All Mathematics lessons →

Example problems

wide ellipse - Semi-axes 5 and 3 put the foci at c = 4: the 3-4-5 triangle sitting inside the ellipse, and an eccentricity of exactly 0.8.
tall ellipse - The same 5-by-3 ellipse as Wide ellipse, stood upright and moved to (-1, 1). Same c = 4, same eccentricity 0.8 - orientation is not shape.
parabola up - Vertex at (0, -2), focus at the origin, directrix at y = -4. The vertex sits 2 from each, which is the entire definition of a parabola.
parabola right - p = 3, so the focus at (1, 0) and the directrix at x = -5 stand 3 either side of the vertex, and the chord through the focus is 4p = 12 across.
hyperbola x - a = 4 and b = 2 give c = 4.472136 and asymptotes of slope 0.5. Eccentricity 1.118034, only just past the parabola's 1.
hyperbola y - a = 3 and b = 4 give c = 5, the same 3-4-5 triangle as Wide ellipse - except here it is c² = a² + b², not a² = b² + c². Eccentricity 1.666667.