Equation of hyperbola satisfying give conditions: Vertices (0,,pm 5), foci (0,,±8)

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10-2. Parabola, Ellipse, Hyperbola

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

A

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

B

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

C

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

D

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

Solution

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

Standard 11

Mathematics

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