Question 1165474
the equation of the ellipse is:


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

given:

center is at ({{{-3}}}, {{{-1}}})=> {{{h=-3}}}, {{{k=-1}}}

so far we have:

{{{(x+3)^2/a^2 + (y+1)^2/b^2 = 1}}}


focus at ({{{1}}},{{{ -1}}}) 
vertex at ({{{2}}}, {{{-1}}}) 

The major axis a is the segment that contains both foci  and has its endpoints on the ellipse. These endpoints are called the vertices. The {{{midpoint}}} of the major axis is the {{{center}}}  of the ellipse.

so using midpoint formula,  the other vertex will be  at 

{{{(2+x)/2=-3}}}=>{{{ 2+x=-6}}}=>{{{ x=-6-2}}}=> {{{x=-8}}}
{{{(-1+y)/2=-1}}}=>{{{-1+y=-2}}}=>{{{y=-2+1}}}=>{{{y=-1}}}

other vertex will be  at ({{{-8}}},{{{-1}}})


and the major axis ={{{2a}}} is distance between  vertices

{{{2a=sqrt((2-(-8))^2+(-1-(-1))^2)}}}

{{{2a=sqrt(10^2+0^2)}}}
{{{2a=10}}}
{{{a=5}}}


so far 

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

{{{(x+3)^2/25 + (y+1)^2/b^2 = 1}}}

The formula generally associated with the focus of an ellipse is{{{ c^2=a^2-b^2}}} where {{{c }}}is the distance from the focus to center , {{{a }}}is the distance from the center to a vetex and {{{b }}}is the distance from the center to a co-vetex


if center is at ({{{-3}}}, {{{-1}}}) and focus at ({{{1}}}, {{{-1}}}) 

 and {{{c}}} is the distance from the focus to center, then=>

*[invoke formula_distance 1, -1, -3, -1]


so, {{{a=5}}} and{{{c=4}}}

{{{c^2=a^2-b^2}}}
{{{b^2=a^2-c^2}}}
{{{b^2=5^2-4^2}}}
{{{b^2=25-16}}}
{{{b^2=9}}}

so, your equation is

{{{(x+3)^2/25 + (y+1)^2/9 = 1}}}

expand:

{{{(x+3)^2/25 + (y+1)^2/9 = 1}}}... both sides multiply by {{{25*9=225}}}

{{{225(x+3)^2/25 + 225(y+1)^2/9 = 225}}}..........simplify

{{{9(x+3)^2 + 25(y+1)^2 = 225}}}

{{{9(x^2 + 6 x + 9) + 25(y^2+2y+1) = 225}}}

{{{9x^2 + 54 x + 81 + 25y^2+50y+25 = 225}}}

{{{9x^2 + 54 x + 81 + 25y^2+50y+25 = 225}}}

{{{9 x^2 + 54x + 25y^2 + 50y + 106= 225}}}

{{{9 x^2 + 54x + 25y^2 + 50y + 106-225=0}}}

{{{9x^2 + 25y^2 + 54x + 50y - 119 = 0}}}=> option d. 


d. (9x)2 + (25y)2 + 54x + 50y  -  119 = 0 -> should be  9x^2 + 25y^2 + 54x + 50y  -  119 = 0