Question 1168920


 
 The equation of an ellipse can be given as,

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

where

'{{{a}}}' represents the semi-major axis (half of the length of the major axis)
'{{{b}}}' represents the semi-minor axis (half of the length of the minor axis)
‘{{{h}}}’ represents the {{{x }}}coordinate of the center
‘{{{k}}}’ represents the {{{y}}} coordinate of the center


if the endpoints of minor axis are ({{{1}}},{{{3}}}) and ({{{1}}},{{{-1}}}) ,  minor axis length is equal to the distance between them which is {{{4}}}

half of the length of the minor axis is {{{b=4/2=2}}}

=> the center mast be half way between endpoints, and it is at ({{{1}}},{{{1}}})


 a focus at ({{{-1}}},{{{1}}}) 
 
the distance between foci  ({{{-1}}},{{{1}}})  and center ({{{1}}},{{{1}}})  
focus is {{{2 }}}units from the center, so {{{c = 2}}}


using the Pythagorean fact of all ellipses

{{{a^2=c^2+b^2}}}

{{{a^2=2^2+2^2=8}}}


so, your formula is

{{{(x-1)^2/8+(y-1)^2/4=1}}}


{{{ drawing( 600, 600, -10, 10, -10, 10,
circle(1,1,.12), locate(1,1,C(1,1)),
circle(-1,1,.12), locate(-1,1,F(-1,1)),
circle(1,3,.12), locate(1,3,p(1,3)),
circle(1,-1,.12), locate(1,-1,p(1,-1)),
graph( 600, 600, -10, 10, -10, 10, (1/2)(2-sqrt(2)sqrt(-x^2 + 2x + 7)), (1/2) (sqrt(2) sqrt(-x^2+2x+7) + 2))) }}}