The equation of the hyperbola is:
[tex]\frac{x^2}{8} - \frac{y^2}{8} = 1[/tex]
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- Directrices at [tex]x = \pm 2[/tex]
- Foci at [tex]x = \pm 4[/tex]
They are both in the x-axis, and the center is (0,0), which means that the equation is:
[tex]\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1[/tex]
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Directrices at [tex]x = \pm 2[/tex] means that:
[tex]\frac{a^2}{c} = 2[/tex]
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Foci at [tex]x = \pm 4[/tex] means that:
[tex]c^2 = 4^2 = 16 \rightarrow c = 4[/tex]
Thus:
[tex]a^2 = 2c = 2(4) = 8[/tex]
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For b, we have that:
[tex]a^2 + b^2 = c^2[/tex]
[tex]8 + b^2 = 16[/tex]
[tex]b^2 = 8[/tex]
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Thus, since a and b have been found, the equation of the hyperbola is:
[tex]\frac{x^2}{8} - \frac{y^2}{8} = 1[/tex]
A similar problem is given at https://brainly.com/question/20409089
