Coordinate Geometry · Mathematics

Parabola Lab Simulation

Q: What is the condition for a focal chord of a parabola?

Its ends P(t₁) and P(t₂) satisfy t₁t₂ = −1.

Q: Where do perpendicular tangents to a parabola meet?

On the directrix x = −a, since t₁t₂ = −1 gives the meeting point (−a, a(t₁ + t₂)).

Q: How many normals can be drawn to a parabola from a point?

Up to three: their parameters solve at³ + (2a − h)t − k = 0. There are three distinct real normals exactly when 27ak² < 4(h − 2a)³.

Q: What is the shortest focal chord of a parabola?

The latus rectum, of length 4a: a focal chord has length a(t + 1/t)² ≥ 4a.

Twenty-eight interactive modules on y² = 4ax, from the focus-directrix definition to the JEE Advanced special results. Explore, predict, derive and practise with randomised problems.

Interactive virtual lab · no download · works on phones and laptops

Chapter: Conic Sections
For: Class 11 · JEE Main · JEE Advanced
Time: ~30 min

How to use it

1Pick a module from the menu.
2Drag the orange points or type exact values.
3Pan and zoom the board.
4Use Predict, Derive and Practise; Progress tracks your scores.

What to notice

P(t) = (at², 2at): tangent ty = x + at², normal y = −tx + 2at + at³.
Focal chords have t₁t₂ = −1; the shortest is the latus rectum, 4a.
Perpendicular tangents meet on the directrix.
Three normals pass through (h, k) iff 27ak² < 4(h − 2a)³.

For teachers

Teach it in 31 minutes

A lesson plan, questions to put to the class, a quick quiz you can project or print, and the formulas behind Parabola Lab.

1Hook
~2 min
- Drag a point along y² = 4ax and ask: “Is there a point where the distance to the focus always equals the distance to some fixed line?” Show the directrix.
2Demonstrate
~18 min
- Pick a module from the menu.
- Drag the orange points or type exact values.
- Pan and zoom the board.
- Use Predict, Derive and Practise; Progress tracks your scores.
3Pause and ask
~5 min
- Have a student say this in their own words: P(t) = (at², 2at): tangent ty = x + at², normal y = −tx + 2at + at³.
- Have a student say this in their own words: Focal chords have t₁t₂ = −1; the shortest is the latus rectum, 4a.
- Have a student say this in their own words: Perpendicular tangents meet on the directrix.
- Have a student say this in their own words: Three normals pass through (h, k) iff 27ak² < 4(h − 2a)³.
4Check
~6 min
- Run the 6-question quiz below: project it and take answers by show of fingers (1–4), or print it as a worksheet.

About the Parabola Lab simulation

The Parabola Lab is a complete interactive chapter on the parabola y² = 4ax for JEE Main and JEE Advanced, in 28 modules. It starts from the focus–directrix definition, orientations and the latus rectum, and builds up through parametric points, chords, focal chords, tangents, normals, chord of contact, reflection and loci, to the special results examiners love: the three-normal region, co-normal points, the focal-chord circle touching the directrix, and least distances.

Every module puts a live diagram on a coordinate board you can pan and zoom, with draggable points, exact numeric inputs and the equations updating as you move. Each one follows the same flow: explore, predict the result before the board shows it, read the derivation, then practise on randomised problems with hints and worked solutions. Your progress is tracked by module.

Also covers: parabola jee advanced, focal chord, tangent to parabola, normal to parabola, three normals, chord of contact, parametric parabola, y^2=4ax.