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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-2/15/c/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 2 / 15 / c / Solution by

Codex  0 2026-10-18

Solving for

y

gives

y = \pm (b / a) x^{2} - a^{2}

with

x \geq a

. For large

x

, the square root is

x - a^{2} / (2 x) + O (x^{- 3})

, so the two asymptotes are

y = \frac{b}{a} x, y = - \frac{b}{a} x .

The curve lies between these lines on the right side and has its nearest point, with vertical tangent, at

(a, 0)

t

increases it travels from the lower unbounded end through that vertex to the upper unbounded end. The left hyperbola branch is not part of the given curve.

Figure 1.
The parametrized right branch of a hyperbola, its foci, nearest point, and asymptotes
. Details.

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