Question 1081571
Write the equation of a hyperbola with foci at ({{{3}}}, {{{-3}}}) and ({{{3}}}, {{{7}}}) and vertices at ({{{3}}}, {{{-1}}}) and ({{{3}}},{{{ 5}}}). 

determine if it is
horizontal:  {{{(x-h)^2/a^2 -(y-k)^2/b^2 =1}}}
or
vertical:  {{{(y-k)^2/a^2 - (x-h)^2/b^2 =1}}}

To start, let's graph the information we have:


{{{drawing( 600, 600, -10, 10, -10, 10,
circle(3,-3,.12),circle(3,7,.12),circle(3,-1,.12),circle(3,5,.12),
locate(3,-3,f(3,-3)),locate(3,7,f(3,7)),locate(3,-1,v(3,-1))
,locate(3,5,v(3,5)),
 graph( 600, 600, -10, 10, -10, 10, 0)) }}}


so,you have vertical:  

{{{(y-k)^2/a^2 - (x-h)^2/b^2 =1}}}

now
1. Identify the center point ({{{h}}}, {{{k}}})
it is midpoint of line segment:({{{3}}}, {{{-3}}}) and ({{{3}}}, {{{7}}})
 ({{{(3+3)/2}}}, {{{(-3+7)/2}}})=({{{3}}}, {{{2}}})
The center point is ({{{h}}}, {{{k}}})=({{{3}}}, {{{2}}}).
so,{{{h=3}}} and {{{k=2}}}

put it on a graph:

{{{drawing( 600, 600, -10, 10, -10, 10,
circle(3,-3,.12),circle(3,7,.12),circle(3,-1,.12),circle(3,5,.12),circle(3,2,.12),
locate(3,-3,f(3,-3)),locate(3,7,f(3,7)),locate(3,-1,v(3,-1))
,locate(3,5,v(3,5)),locate(3,2,C(3,2)),
 graph( 600, 600, -10, 10, -10, 10, 0)) }}}


2. Identify {{{a}}} and {{{c}}}

To find {{{a}}}, we'll count from the center to either vertex; {{{a = 3}}}. 
To find {{{c}}}, we'll count from the center to either focus;{{{ c = 5}}}


3. Use the formula {{{c^2 = a^2 + b^2}}} to find {{{b}}} (or {{{b^2}}})

{{{b^2=c^2 -a^2 }}}
{{{b^2=5^2 -3^2 }}}
{{{b^2=25 -9 }}}
{{{b^2=16 }}}->{{{b=4}}}

4. Plug {{{h=3}}}, {{{k=2}}}, {{{a=3}}}, and {{{b=4}}} into the correct pattern.

{{{(y-2)^2/3^2 - (x-3)^2/4^2 =1}}}

{{{(y-2)^2/9 - (x-3)^2/16 =1}}}



{{{drawing( 600, 600, -20, 20, -20, 20,
circle(3,-3,.12),circle(3,7,.12),circle(3,-1,.12),circle(3,5,.12),circle(3,2,.12),
locate(3,-3,f(3,-3)),locate(3,7,f(3,7)),locate(3,-1,v(3,-1))
,locate(3,5,v(3,5)),locate(3,2,C(3,2)),
 graph( 600, 600, -20, 20, -20, 20,sqrt(9(x-3)^2/16 +9)+2,-sqrt(9(x-3)^2/16 +9)+2)) }}}


next problem is of same type; so,I am pretty sure you can do it following these steps