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?

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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?

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Answer 1 · 2015-05-30T22:58:43

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 $7x^{2}-y^{2}=56$ .

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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?

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