hyperbolas

1. foci (+or- 17, 0) vertices (+or- 8, 0)

This hyperbola  will  intercept the x axis  and be centered at (0,0)....so it will have the form

 x^2            y^2

 ___   -      ___      =  1

 a^2           b^2

a = 8

c  = 17

b  =√ [ 17^2  - 8^2]  = √ [289 - 64]  = √ 225  =  15

So....the equation becomes

x^2        -     y^2

___            ____   =  1

64              225

Here's a graph :  https://www.desmos.com/calculator/a2uusijq3k

2. foci (0, +or- 25) vertices (0, +or- 7)

This hyperbola intercepts the y  axis and is centered at (0,0)

The form is

y^2     -  x^2

___       ___    = 1

a^2        b^2

a  = 7

c  = 25

b = √[ 25^2  - 7^2 ]  = √  [625 - 49 ]  =√576  =  24

So.....the equation is :

y^2   -    x^2

___       ___     = 1

49         576

Here's a graph  :   https://www.desmos.com/calculator/fyyifnyzze

3. Find an equation that models the hyperbolic path of a spacecraft around a planet if a=107124 km and c=213125.9 km

Let's let the center be (0,0)  and let  the hyperbola intrcept the y axis

The form is

y^2     -    x^2

___        ____    =    1

a^2         b^2

a = 107124

c = 213125.9

So

b^2  = [213125.9^2  - 107124^2]    .....this is a huge number...let's just leave it as this

So.....the equation is

    y^2             -             x ^2

 ________           ____________________   =     1

 107124^2          [213125.9^2  - 107124^2 ]