Find the equation of the hyperbola satisfying the give conditions: Vertices $(0,\,\pm 3),$ foci $(0,\,±5)$
Find the equation of the hyperbola satisfying the give conditions: Vertices $(0,\,\pm 3),$ foci $(0,\,±5)$
Vertices $(0,\,\pm 3),$ foci $(0,\,±5)$
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 $(0,\,\pm 3), a=3$
since the foci are $(0,\,\pm 5), c=5$
We know that $a^{2}+b^{2}=c^{2}$
$\therefore 3^{2}+b^{2}=52$
$\Rightarrow b^{2}=25-9=16$
Thus, the equation of the hyperbola is $\frac{y^{2}}{9}-\frac{x^{2}}{16}=1$
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