Find the equation of the hyperbola satisfying the give conditions: Vertices $(0,\,\pm 5),$ foci $(0,\,±8)$
Find the equation of the hyperbola satisfying the give conditions: Vertices $(0,\,\pm 5),$ foci $(0,\,±8)$
Vertices $(0,\,\pm 5),$ foci $(0,\,±8) $
Here, the vertices are on the $y-$ axis.
Therefore, the equation of the hyperbola is of the form $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$
since the vertices are $(0,\,\pm 5), \,\,a=5$
since the foci are $(0,\,\pm 8),\,\, c=8$
We know that $a^{2}+b^{2}=c^{2}$
$\therefore $ $5^{2}+b^{2}=8^{2}$
$b^{2}=64-25=39$
Thus, the equation of the hyperbola is $\frac{y^{2}}{25}-\frac{x^{2}}{39}=1$
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