A student asserts that “the larger the eccentricity of a hyperbola, the more ‘open’ its branches appear.” Study the following statements about eccentricity and the shape of a hyperbola centred at the origin.
(i) Explain, with reasons, why a hyperbola must have eccentricity greater than 1.
(ii) Describe qualitatively what happens to the branches of a hyperbola as its eccentricity approaches 1 from above. Comment on the distance of the vertices from the foci in this limiting case.
(iii) Two horizontal hyperbolas have the same transverse axis length 2a but eccentricities e1 and e2 with e1>e2>1. Without performing any numerical calculation, justify which hyperbola has the smaller angle between its asymptotes and which looks more ‘open’.