Question

How do you write an equation of a hyperbola given the foci of the hyperbola are (8 , 0) and (−8 , 0), and the asymptotes are y =sqrt(7)x and y =−sqrt(7)x?

Answer 1

In many textbooks, standard notation for such a hyperbola is `(x/a)^2-(y/b)^2=1`. For such a hyperbola, the asymptotes are `y=\pm \frac{b}{a}x`. Hence, `b/a=\sqrt{7}`. Also for such a hyperbola, the foci are at the points whose `x` coordinates are `x=\pm\sqrt{a^{2}+b^{2}}`, implying that `\sqrt{a^{2}+b^{2}}=8` so that `a^{2}+b^{2}=64`.

Subtituting `b=a\sqrt{7}` into this last equation gives `a^{2}+7a^{2}=64`, or `a^{2}=8` so that `a=\sqrt{8}=2\sqrt{2}`. This then implies that `b=2\sqrt{2}\sqrt{7}=2\sqrt{14}`.

The equation of the hyperbola is `(x/(2\sqrt{2}))^{2}-(y/(2\sqrt{14}))^{2}=1`, which can also be written as `x^{2}/8-y^{2}/56=1` or `7x^{2}-y^{2}=56`.