Question 1194464
The equation of a hyperbola is 

{{{(x-h)^2/a^2-(y-k)^2/b^2=1}}}, where ({{{h}}},{{{k}}}) is the center, {{{a}}} and {{{b}}} are the lengths of the semi-major and the semi-minor axes.

given:

center at ​({{{4}}}​,{{{-6}}}​)=>{{{h=4}}}, {{{k = -6}}}

focus at ​({{{7}}}​,{{{-6​}}}) =({{{h+c}}},{{{k}}})=>{{{h=4}}}, {{{4+c=7}}}=>{{{c=7-4=3}}}

vertex at ({{{6}}},{{{-6}}})=({{{h+a}}},{{{k}}})=>{{{h=4}}},{{{4+a=6}}}=>{{{a=2}}}

{{{b^2=c^2-a^2}}}
 {{{b^2=3^2-2^2}}}
 {{{b^2=9-4}}}
 {{{b^2=5}}}
{{{b=sqrt(5)}}}

The equation of a hyperbola is 

{{{(x-4)^2/2^2-(y-(-6))^2/(sqrt(5))^2=1}}}

{{{(x-4)^2/4-(y+6)^2/5=1}}}


{{{ drawing( 600, 600, -10, 10, -10, 10,
circle(4,-6,.12), locate(4,-6,C(4,-6)),
circle(7,-6,.12), locate(7,-5.5,F(7,-6)),
circle(6,-6,.12), locate(6,-6,V(6,-6)),
graph( 600, 600, -10, 10, -10, 10, (1/2)(-sqrt(5)*sqrt(x^2-8x+12)-12), (1/2) (sqrt(5) *sqrt(x^2- 8x+12) -12))) }}}