Question 1181166
Given by the equation 

{{{49x^2 + 9y^2 = 441}}}........both sides divide by {{{441}}}

{{{49x^2/441 + 9y^2/441 = 441/441}}}

{{{x^2/9 + y^2/49 = 1}}}

=>center is at origin, {{{h=0}}},{{{k=0}}}
=>{{{a=sqrt(9)=3}}}
=>{{{b=sqrt(49)=7}}}

{{{b>a}}}=> you have vertical ellipse

{{{c=sqrt(b^2-a^2)}}}
{{{c=sqrt(49-9)}}}
{{{c=sqrt(40)}}}
{{{c=sqrt(4*10)}}}
{{{c=2sqrt(10)}}}

1. Center
2. Foci: ({{{h}}},{{{c}}}),({{{h}}},{{{-c}}})


𝐹1 ({{{0}}}, {{{2sqrt(10)}}})
𝐹2 ({{{0}}}, {{{-2sqrt(10)}}})

3. Vertices: ({{{h}}},{{{b}}}),({{{h}}},{{{-b}}})
𝑉1 ({{{0}}},{{{7}}})
𝑉2 ({{{0}}},{{{-7}}})

4. Co-vertices:({{{a}}},{{{k}}}),({{{-a}}},{{{k}}})
𝐵1 ({{{3}}},{{{0}}})
𝐵2 ({{{-3}}},{{{0}}})


5. Endpoints of Latus Rectum

use the coordinate of the focus {{{y=2sqrt(10)}}}, substitute in ellipe equation and solve for {{{x}}}
{{{x^2/9 + (2sqrt(10))^2/49 = 1}}}
{{{x^2/9 + 40/49 = 1}}}
{{{x^2/9   = 1-40/49}}}
{{{x^2/9   = 9/49}}}
{{{x^2  = 9(9/49)}}}...simplify
{{{x^2  = 9/7}}}

{{{x= sqrt(9/7)}}}
{{{x= 3/sqrt(7)}}}
{{{x= 3sqrt(7)/7}}} or {{{x= -3sqrt(7)/7}}}

,

𝐸1({{{3sqrt(7)/7}}},{{{2sqrt(10)}}})
𝐸2({{{-3sqrt(7)/7}}},{{{2sqrt(10)}}})
𝐸3({{{3sqrt(7)/7}}},{{{-2sqrt(10)}}})
𝐸4({{{-3sqrt(7)/7}}},{{{-2sqrt(10)}}})

6. Directrices:  {{{y=49sqrt(10)/20}}} , {{{y=-49sqrt(10)/20}}}
7. Eccentricity:{{{2sqrt(10)/7}}}
8. Length of LR:{{{18/7}}}
9. Length of Major Axis:{{{14}}}
10.Length of Minor Axis:{{{6}}}


{{{drawing ( 600, 600, -10, 10, -10, 10,
circle(0,0,.13),locate(0.3,0.5,C(0,0)),
circle(0,7,.13),locate(0.3,7.3,V),
circle(0,-7,.13),locate(0.3,-7,V),

circle(3,0,.13),locate(3,0.5,cV),
circle(-3,0,.13),locate(-3,0.5,cV),

circle(0,-2sqrt(10),.13),locate(0.3,-2sqrt(10),F),
circle(0,2sqrt(10),.13),locate(0.3,2sqrt(10),F),
circle(3sqrt(7)/7,2sqrt(10),.13),locate(3sqrt(7)/7,2sqrt(10),E1),
circle(-3sqrt(7)/7,2sqrt(10),.13),locate(-3sqrt(7)/7,2sqrt(10),E2),

circle(3sqrt(7)/7,-2sqrt(10),.13),locate(3sqrt(7)/7,-2sqrt(10),E3),
circle(-3sqrt(7)/7,-2sqrt(10),.13),locate(-3sqrt(7)/7,-2sqrt(10),E4),

blue(line(3sqrt(7)/7,2sqrt(10),-3sqrt(7)/7,2sqrt(10))), blue(line(3sqrt(7)/7,-2sqrt(10),-3sqrt(7)/7,-2sqrt(10))),
graph( 600, 600, -10, 10, -10, 10,- sqrt(49(1-x^2/9 )),sqrt(49(1-x^2/9 )) ,-49sqrt(10)/20,49sqrt(10)/20)) }}}