Simulation · Maths · Class 11
A hyperbola keeps a constant difference
From the lesson Hyperbola: foci and axes in Conic Sections. Change the values and watch what happens.
A hyperbola keeps a constant differenceMaths · Class 11
The idea behind it
NCERT §10.6, §10.6.1
- A hyperbola is the set of all points of a plane for which the difference of the distances from two fixed points, the foci, is a constant. 'Difference' means the distance to the farther focus minus the distance to the nearer one.
- The mid-point of the foci is the centre. The line through the foci is the transverse axis and the line through the centre perpendicular to it is the conjugate axis. The hyperbola meets the transverse axis at its two vertices.
- The distance between the foci is 2c and between the vertices 2a (the length of the transverse axis). The quantity b is defined by b = √(c² − a²), and 2b is the length of the conjugate axis.
- Taking P at a vertex shows that the constant difference is exactly 2a, the distance between the vertices.
- Eccentricity e = c/a, as for the ellipse. Since c is at least a, e is never less than 1, and each focus is ae from the centre.
- Compare the ellipse: there c² = a² − b² and e < 1; for the hyperbola c² = a² + b² and e > 1. Mixing these two relations is the commonest slip in this chapter.