Find the equation of the hyperbola satisfying | vedclass

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Question

(A) The vertices are $(0, \pm 3)$,which implies the hyperbola is along the $y$-axis.The standard equation is $\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}

Vedclass · Accepted Answer

$\frac{y^{2}}{9} - \frac{x^{2}}{16} = 1$

Vedclass · Answer

(A) The vertices are $(0, \pm 3)$,which implies the hyperbola is along the $y$-axis.The standard equation is $\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1$.Given vertices $(0, \pm a) = (0, \pm 3)$,we have $a = 3$,so $a^{2} = 9$.Given foci $(0, \pm c) = (0, \pm 5)$,we have $c = 5$,so $c^{2} = 25$.For a hyperbola,$c^{2} = a^{2} + b^{2}$.Substituting the values: $25 = 9 + b^{2}$.$b^{2} = 25 - 9 = 16$.Thus,the equation is $\frac{y^{2}}{9} - \frac{x^{2}}{16} = 1$.

Find the equation of the hyperbola satisfying the given conditions: Vertices $(0, \pm 3)$,foci $(0, \pm 5)$.

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