Question 759881
what is the standard form of the equation of a hyperbola with vertex (0,4) focus at (0,5) and center at (0,1)?
<pre>
Hyperbolas have the equation 

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

if they look like this: <font color="red">)(</font>

and the equation

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

if they look like this:{{{drawing(20,25,8,12,8,12, graph(20,25,8,12,8,12,sqrt(1+(x-10)^2)+10),graph(20,25,8,12,8,12,-sqrt(1+(x-10)^2)+10) )}}}

We plot the three given points for the vertex V, focus F 
and center C:

{{{drawing(400,400,-9,9,-8,10, grid(1),circle(0,1,.1),circle(0,1,.15),circle(0,1,.08),circle(0,1,.04), 
circle(0,4,.15),circle(0,4,.08),circle(0,4,.04),circle(0,4,.1), 
circle(0,5,.15),circle(0,5,.08),circle(0,5,.04),circle(0,5,.1),
locate(0.3,4.7,V),locate(0.3,5.7,F),locate(0.3,1.7,C)


 )}}}

So it looks like this:{{{drawing(20,25,8,12,8,12, graph(20,25,8,12,8,12,sqrt(1+(x-10)^2)+10),graph(20,25,8,12,8,12,-sqrt(1+(x-10)^2)+10) )}}}

and has the equation

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

We know that the center (h,k) is (0,1). We know that 
the semi-transverse axis, a, is the distance from the 
center to a vertex, and it is 3 units from C to V, so
a=3.  There is another vertex 3 units below the center
at (0,-2).

So we now have everything but b:

{{{(y-1)^2/3^2}}}{{{""-""}}}{{{(x-0)^2/b^2}}}{{{""=""}}}{{{1}}}

We know that c is the distance from the center to a 
focus, and there are 4 units from C to F so c=4.  
There is another focus 4 units below the center at 
(0,-3).

In all hyperbolas we have the Pythagorean property

c² = a² + b²
4² = 3² + b²
16 = 9 + b²
 7 = b²
&#8730;<span style="text-decoration: overline">7</span> = b

So now we know that b² = 7, a² =3² = 9, so the equation is:

{{{(y-1)^2/9}}}{{{""-""}}}{{{(x-0)^2/7}}}{{{""=""}}}{{{1}}}

or change (x-0)² to just x²

{{{(y-1)^2/9}}}{{{""-""}}}{{{x^2/7}}}{{{""=""}}}{{{1}}}

Here is the complete graph.  The conjugate axis is the horizontal
line through the center, the width of the defining rectangle.
It is 2b units wide or 2&#8730;<span style="text-decoration: overline">7</span>, &#8730;<span style="text-decoration: overline">7</span> on each side or about 2.7 on each 
side of the center.
  

{{{drawing(400,400,-9,9,-8,10, circle(0,1,.1),circle(0,1,.15),
green(line(20,-21.67786838,-20,23.67786838),line(-20,-21.67786838,20,23.67786838)),blue(line(-sqrt(7),1,sqrt(7),1)),
graph(400,400,-9,9,-8,10,1+sqrt((9x^2+63)/7)),
graph(400,400,-9,9,-8,10,1-sqrt((9x^2+63)/7)),

circle(0,1,.08),circle(0,1,.04),grid(1), 
circle(0,4,.15),circle(0,4,.08),circle(0,4,.04),circle(0,4,.1), 
circle(0,5,.15),circle(0,5,.08),circle(0,5,.04),circle(0,5,.1),
locate(0.3,4.7,V),locate(0.3,5.7,F),locate(0.3,1.7,C),
graph(400,400,-9,9,-8,10),rectangle(-sqrt(7),-2,sqrt(7),4),line(-sqrt(7),-2,sqrt(7),-2),line(-sqrt(7),4,sqrt(7),4)






 )}}}

Edwin</pre>