Question 1181119
 the equation of the ellipse

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

given:
center at ({{{2}}}, {{{3}}}), 

since center is at ({{{h}}}, {{{k}}})=>{{{h=2}}}, {{{k=3}}}
vertices at ({{{2}}}, {{{9}}}) and ({{{2}}}, {{{-3}}}),

since vertices at ({{{h}}}, {{{k+b}}}), ({{{h}}}, {{{k-b}}})
{{{3+b=9}}}=>{{{b=6}}}
 
=> since {{{x}}} coordinates same, major axis parallel {{{y}}} axis, and you have vertical ellipse

 eccentricity of {{{2/3}}}

{{{e=sqrt(b^2-a^2)/b }}}
{{{2/3=sqrt(6^2-a^2)/6}}}
{{{(2/3)^2=(36-a^2)/6^2}}}
{{{(4/9)36=(36-a^2)}}}
{{{(4/1)4=36-a^2}}}
{{{16=36-a^2}}}
{{{a^2=36-16}}}
{{{a^2=20}}}
{{{a=sqrt(20)}}}
{{{a=sqrt(4*5)}}}
{{{a=2sqrt(5)}}}

{{{c=sqrt(b^2-a^2)}}}
{{{c=sqrt(36+20)}}}
{{{c=sqrt(16)}}}
{{{c=4}}}


so, your equation is

{{{(x-2)^2/20+(y-3)^2/36=1}}}





1. Center:({{{2}}}, {{{3}}})
2. Foci:  ({{{h}}}, {{{k+c}}} ), ({{{h}}},{{{ k-c}}} )
𝐹1 ({{{2}}}, {{{3+4}}} ) =>({{{2}}}, {{{7}}} )
𝐹2 ({{{2}}}, {{{3-4}}} ) =>({{{2}}}, {{{-1}}} )

3. Vertices: ({{{h}}}, {{{k+b}}} ), ({{{h}}},{{{ k-b}}} )

𝑉1 ({{{2}}}, {{{3+6}}} )  =>({{{2}}}, {{{9}}} )
𝑉2({{{2}}},{{{ 3-6}}} )  =>({{{2}}}, {{{-3}}} )

4. Co-vertices ({{{h+a}}}, {{{k}}} ), ({{{h-a}}},{{{ k}}} )

𝐵1({{{2+2sqrt(5)}}}, {{{3}}} )=>({{{6.5}}}, {{{3}}} )

𝐵2({{{2-2sqrt(5)}}}, {{{3}}} )=>({{{-2.5}}}, {{{3}}} )

5. Endpoints of Latus Rectum

foci is ({{{2}}}, {{{7}}} ), endpoints E1 and E2 lie on a line {{{y=7}}}

substitute in equation of ellipse to calculate {{{x}}} coordinates

{{{(x-2)^2/20+(7-3)^2/36=1}}}...solve it and you will get

{{{x = -4/3}}} and {{{x = 16/3}}}

so, 
E1 ({{{-4/3}}}, {{{7}}})

E2({{{16/3}}}, {{{7}}})

other focus is at ({{{2}}}, {{{-1}}} ),endpoints Ee and E4 lie on a line {{{y=-1}}}

{{{x = -4/3}}} and {{{x = 16/3}}}


E3 =>({{{-4/3}}}, {{{-1}}})

E4 =>({{{16/3}}},{{{-1}}})

6. Directrices: {{{y=-6}}}, {{{y=12}}}}

7. Eccentricity:{{{2/3}}}

8. Length of LR:   {{{a^2/k=20/3}}}≈ {{{6.66666666666667}}}

9. Length of Major Axis:{{{12}}}

10.Length of Minor Axis:{{{2*2sqrt(5)=4sqrt(5)=8.9}}}


{{{drawing ( 600, 600, -15, 15, -15, 15,
circle(2, 3,.15),locate(2, 3,C),
circle(2, 7,.15),locate(2, 7,F),
circle(2, -1,.15),locate(2, -1,F),
circle(2, 9,.15),locate(2, 9,V),
circle(2, -3,.15),locate(2, -3,V),
circle(6.5, 3,.15),locate(6.5, 3,cV),
circle(-2.5, 3,.15),locate(-2.5, 3,cV),
circle(-4/3, 7,.15),locate(-4/3, 7,E1),
circle(16/3, 7,.15),locate(16/3, 7,E2),
circle(-4/3, -1,.15),locate(-4/3, -1,E3),
circle(116/3, -1,.15),locate(16/3, -1,E4),
graph( 600, 600, -15, 15, -15, 15,-sqrt(36(1-(x-2)^2/20))+3,sqrt(36(1-(x-2)^2/20))+3 )) }}}