Hyperbola with a=6, b=8: find c and e

Given that the real semi-axis of a hyperbola is 6 and the imaginary semi-axis is 8, find its semi-focal distance and eccentricity.

Step-by-step

1. Hyperbola: a = 6, b = 8
   The real semi-axis and imaginary semi-axis of a hyperbola determine its shape.
2. c² = a² + b² = 6² + 8² = 100 ⇒ c = 10
   In a hyperbola, c is the largest, so we compute a² plus b².
3. e = c / a = 10 / 6 = 1.666667
   Eccentricity measures how wide it opens; a hyperbola has eccentricity greater than 1
4. Asymptotes y = ±(b/a)x = ±1.333333x
   The lines that the two branches of a hyperbola approach infinitely closely without intersecting are its asymptotes.
5. Back-substitution check: c² = a² + b² ⇒ 100 = 36 + 64
   Use the relationship among a, b, and c to back-check that the semi-focal distance is correct.

Why it works

The difference between the hyperbola and ellipse formulas lies in that middle sign: for an ellipse, the semi-major axis is largest so you subtract; for a hyperbola, the semi-focal distance is largest so you add. Confusing these two signs is the most typical mistake in this chapter. Eccentricity in a hyperbola must be greater than one, which you can use to self-check: if your result is less than one, you definitely used the ellipse formula by mistake.

Common mistakes

• Writing the hyperbola formula as semi-major axis squared minus semi-minor axis squared.
• Getting an eccentricity less than one without checking whether the wrong formula was used.
• Swapping the real and imaginary semi-axes when substituting, which still yields a mathematically valid number and is hard to notice.

Related

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