Question 1190959
 
given:

Center at ({{{0}}},{{{0}}}) ->{{{h=0}}} and {{{k=0}}}
Vertex ({{{0}}},{{{-6}}})  => the other vertex must be at ({{{0}}},{{{6}}})  ->shows common {{{x}}} coordinate {{{0}}}. So {{{y}}} axis is major axis and {{{x}}} axis is minor axis, therefore {{{b}}} greater than {{{a}}}

{{{2b}}}  is the distance between vertices and centre=>{{{2b=12}}}->{{{b=6}}}

 end of the minor axis ({{{4}}},{{{0}}}) -> {{{a=4}}}

Standard equation for this ellipse is

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

your equation is:

{{{(x - 0)^2/4^2 + (y - 0)^2 /6^2= 1}}}

{{{x ^2/16 + y ^2 /36= 1}}}


{{{ drawing(600, 600, -10, 10, -10, 10,
circle(0,0,.12), locate(0.2,0.5,C(0,0)),
circle(0,-6,.12), locate(0,-6,V(0,-6)),
circle(0,6,.12), locate(0,6,V(0,6)),
graph(600, 600, -10, 10, -10, 10,  -(3/2)sqrt(16 - x^2),  (3sqrt(16 - x^2))/2)) }}}