Lesson 9 of 12 · 6 min
Hyperbola: foci and axes
NCERT §10.6, §10.6.1
Two club members stand 10 m apart on the terrace, at F₁(−5, 0) and F₂(5, 0). A clap somewhere reaches one of them earlier, and the gap in arrival time says the clap was always 6 m nearer to F₂ than to F₁. Where could the clapper be?
The lesson in notes
In short
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.