Find the equation of the hyperbola satisfying | vedclass

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Question

(A) The vertices are $(0, \pm 5)$,which lie on the $y$-axis. Thus,the equation of the hyperbola is of the form $\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{

Vedclass · Accepted Answer

$\frac{y^{2}}{25} - \frac{x^{2}}{39} = 1$

Vedclass · Answer

(A) The vertices are $(0, \pm 5)$,which lie on the $y$-axis. Thus,the equation of the hyperbola is of the form $\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1$. Given the vertices $(0, \pm a) = (0, \pm 5)$,we have $a = 5$,so $a^{2} = 25$. Given the foci $(0, \pm c) = (0, \pm 8)$,we have $c = 8$,so $c^{2} = 64$. For a hyperbola,we know the relation $c^{2} = a^{2} + b^{2}$. Substituting the values,$64 = 25 + b^{2}$,which gives $b^{2} = 64 - 25 = 39$. Substituting $a^{2}$ and $b^{2}$ into the standard form,the equation is $\frac{y^{2}}{25} - \frac{x^{2}}{39} = 1$.

Find the equation of the hyperbola satisfying the given conditions: Vertices $(0, \pm 5)$,foci $(0, \pm 8)$.

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