A string, two pins and an ellipse
"A string, two pins and an ellipse" is a free, interactive Class 11 Maths simulation for JEE, built on the lesson "Ellipse: foci and axes" in Conic Sections. A string 10 units long is pinned at the two foci and held tight by the pencil P. Move P round: PF₁ changes, PF₂ changes, but their sum is always 10 = 2a. An ellipse is the set of points with PF₁ + PF₂ = 2a. The foci are c from the centre, b² = a² − c², and e = c/a runs from 0 (a circle) towards 1 (very flat). JEE Main asked 116 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
From the lesson Ellipse: foci and axes in Conic Sections. Change the values and watch what happens.
In JEE Main
42 shifts analysed
116 questions from Conic Sections across the 42 JEE Main shifts Lumi has analysed (2017–2026).
The idea behind it
NCERT §10.5, §10.5.1, §10.5.2
- An ellipse is the set of all points of a plane whose distances from two fixed points, the foci, add up to the same constant. That constant must be greater than the distance between the foci.
- The mid-point of the foci is the centre. The major axis is the segment through the foci; the minor axis is the segment through the centre perpendicular to it. The ends of the major axis are the vertices.
- Write 2a for the length of the major axis, 2b for the minor axis and 2c for the distance between the foci, so a and b are the semi-major and semi-minor axes and each focus is c from the centre.
- At a vertex P, PF₁ + PF₂ = (a + c) + (a − c) = 2a, so the constant sum is always 2a. At an end Q of the minor axis both distances are √(b² + c²), so 2√(b² + c²) = 2a.
- Hence a² = b² + c², or c = √(a² − b²). The focus, the centre and an end of the minor axis form a right triangle with hypotenuse a.
- The eccentricity of an ellipse is e = c/a, the ratio of the centre-to-focus distance to the centre-to-vertex distance. Each focus is therefore ae from the centre.
- Since c < a, e < 1. As c shrinks towards 0 the foci merge, b approaches a and the ellipse rounds off towards a circle; as c approaches a it flattens.
Questions
3 answers
- What does the "A string, two pins and an ellipse" simulation show?
- A string 10 units long is pinned at the two foci and held tight by the pencil P. Move P round: PF₁ changes, PF₂ changes, but their sum is always 10 = 2a.
- 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 "Ellipse: foci and axes" in Conic Sections (Class 11 Maths).
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