I am having issues with finding hyperbola equations with a translation when it gives the vertices and foci.
vertices: (-5,5) (5,5)
foci: (-7,5) (7,5)
The focal axis is horizontal, so the equation is in the form
$\displaystyle \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$
The center (h, k) is in between the two vertices, so it's (0, 5).
a is the distance from the center to one of the vertices, so a = 5.
c is the distance from the center to one of the foci, so c = 7.
The relationship between a, b, c is
$\displaystyle c^2 = a^2 + b^2$
so solve for b:
$\displaystyle 7^2 = 5^2 + b^2$
$\displaystyle 49 = 25 + b^2$
$\displaystyle 24 = b^2$
$\displaystyle b = \sqrt{24}$
So the equation should be
$\displaystyle \frac{x^2}{25} - \frac{(y - 5)^2}{24} = 1$
Nothing to it!
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