Conic Sections

Maths · Class 11

Simulation · Maths · Class 11

Focus, directrix and the parabola

"Focus, directrix and the parabola" is a free, interactive Class 11 Maths simulation for JEE, built on the lesson "Parabola and its standard forms" in Conic Sections. The blue curve is y² = 4ax, with focus F(a, 0) and directrix x = −a. Slide P along the curve: the orange PF and the green PB (straight across to the directrix) stay equal at every point. Every point of y² = 4ax is as far from F(a, 0) as from the line x = −a. The vertex sits halfway between them, and the latus rectum through the focus has length 4a. JEE Main asked 116 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).

From the lesson Parabola and its standard forms in Conic Sections. Change the values and watch what happens.

Focus, directrix and the parabolaMaths · Class 11

In JEE Main

42 shifts analysed

116 questions from Conic Sections across the 42 JEE Main shifts Lumi has analysed (2017–2026).

See the past questions

The idea behind it

NCERT §10.4, §10.4.1

  • A parabola is the set of all points of a plane that are equally far from a fixed line, the directrix, and a fixed point not on it, the focus. The name is often read as 'para' (for) and 'bola' (throwing): the path of a thrown ball.
  • If the fixed point lay on the fixed line, the equidistant points would form the straight line through that point perpendicular to the fixed line: a degenerate parabola.
  • The axis is the line through the focus perpendicular to the directrix. The vertex is where the parabola meets its axis, midway between the focus and the directrix.
  • Put the vertex at the origin and the focus at (a, 0) with a > 0, so the directrix is x = −a. For P(x, y), PF² = (x − a)² + y² and the distance to the directrix is |x + a|. Setting them equal and squaring gives y² = 4ax, and every point of y² = 4ax satisfies PF = x + a in return.
  • The four standard forms: y² = 4ax (opens right), y² = −4ax (opens left), x² = 4ay (opens up), x² = −4ay (opens down), with a > 0 in each. In every one the vertex is the origin and the focus lies on an axis at distance a.
  • Reading a standard equation: a y² term means the axis of symmetry is the x-axis; an x² term means it is the y-axis. The sign of the first-degree term gives the direction of opening.
  • In y² = 4ax, x can never be negative, so the curve lies in the first and fourth quadrants and runs on without end, symmetric about the x-axis.
  • Parabolas with a vertex elsewhere, or an axis not along a coordinate axis, are outside this chapter.

Questions

3 answers

What does the "Focus, directrix and the parabola" simulation show?
The blue curve is y² = 4ax, with focus F(a, 0) and directrix x = −a. Slide P along the curve: the orange PF and the green PB (straight across to the directrix) stay equal at every point.
Does JEE Main ask questions from Conic Sections?
Yes. JEE Main asked 116 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
Is this simulation free?
Yes. It is free and runs in the browser on a phone or a laptop. It belongs to the lesson "Parabola and its standard forms" in Conic Sections (Class 11 Maths).

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