Question 953443
Standard form of hyperbola with {{{horizontal }}}{{{transverse}}} axis:
{{{ (x-h)^2/a^2-(y-k)^2/b^2=1}}}, with ({{{h}}},{{{k}}}) being the ({{{x}}},{{{y}}}) coordinates of the center.
Standard form of hyperbola with {{{vertical}}}{{{ transverse}}} axis: 
{{{(y-k)^2/a^2-(x-h)^2/b^2=1}}}, with ({{{h}}},{{{k}}}) being the ({{{x}}},{{{y}}}) coordinates of the center.

The difference between these two forms is that the {{{(x-h)^2}}} and {{{(y-k)^2}}} terms are interchanged.

Given equation of hyperbola:

{{{ (y-3)^2/4^ 2 - (x+5)^2 / 2^ 2=1}}}

This is a hyperbola of the 2nd form listed. It has a vertical transverse axis with center at (3,-5).
{{{a^2=4^2}}}
{{{a=4}}}=>distance from center to vertex

Length of transverse axis is the distance between vertices; so, =>{{{2a=8}}}
See the graph below for visual evidence of the algebra above:



{{{drawing( 600, 600, -10, 10, -10, 10,
circle(-5,-1,.12),circle(-5,7,.12),line(-5,-1,-5,7),
 graph( 600, 600, -10, 10, -10, 10, sqrt(16((x+5)^2 / 2^ 2+1))+3,-sqrt(16((x+5)^2 / 2^ 2+1))+3)) }}}