Question

guided practice
now solve a similar problem on your own.
find an equation of a hyperbola given that ( b = 8 ) and the coordinates of the foci are ( (0, pm 10) ). assume that the center of the hyperbola is at the origin and that the transverse axis is vertical.
type your answer and then click or tap done.
( \frac{y^2}{ square } - \frac{x^2}{ square } = 1 )

Explanation:

Step1: Identify hyperbola parameters

For a vertical transverse axis hyperbola centered at the origin, the standard form is $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$, with foci at $(0, \pm c)$. We know $b=8$, $c=10$.

Step2: Calculate $a^2$ via hyperbola relation

Use $c^2 = a^2 + b^2$. Rearrange to $a^2 = c^2 - b^2$.
$a^2 = 10^2 - 8^2 = 100 - 64 = 36$

Step3: Calculate $b^2$

$b^2 = 8^2 = 64$

Step4: Substitute into standard equation

Substitute $a^2=36$ and $b^2=64$ into the standard form: $\frac{y^2}{36} - \frac{x^2}{64} = 1$

Answer:

$\frac{y^2}{36} - \frac{x^2}{64} = 1$