Question 1065504
The major axis,
containing center (4,2) and focus (7,2),
is part of line {{{y=2}}} .
The focal distance is the distance between center and focus,
{{{c=7-4=3}}} .
There is another focus on the major axis,
with {{{system(x=4-3=1,y=2)}}} .
The ellipse looks like this:
{{{drawing(300,225,-1,9,-2,5.5,
grid(1),red(circle(4,2,0.1)),
red(circle(7,2,0.1)),red(circle(1,2,0.1)),
red(arc(4,2,8,2sqrt(7),0,360))
)}}} It is tangent to the y-axis
at vertex (0,2) ,
because an ellipse with a horizontal major axis
can only be tangent to a vertical line at a vertex.
So,the semi-major axis length is
the distance from that vertex to the center,
{{{a=4-0=4}}} .
Ŵe know that the length of the semi-minor axis, {{{b}}} ,
is related to {{{a}}} and {{{c}}} by
{{{b^2+c^2=A^2}}} .
So, {{{b^2+3^2=4^2}}}
{{{b^2+9=16}}}
{{{b^2=16-9}}}
{{{b^2=7}}}
{{{b=sqrt(7)}}} .
So, the length of the minor axis is
{{{highlight(2sqrt(7))}}} .