Question 1181167
{{{9x^2 + 16y^2 -54x - 64y + 1 = 0}}}

{{{(9x^2-54x) + (16y^2  - 64y) =-1}}}

{{{9(x^2-6x) +16 (y^2  - 4y) =-1}}}

{{{9(x^2-6x+b^2) -9b^2+16 (y^2  - 4y+b^2) -16b^2=-1}}}

{{{9(x^2-6x+3^2) -9*3^2+16 (y^2  - 4y+2^2) -16*2^2=-1}}}

{{{9(x-3)^2 -81+16 (y - 2)^2 -64=-1}}}

{{{9(x-3)^2 +16 (y - 2)^2 -145=-1}}}

{{{9(x-3)^2 +16 (y - 2)^2 =145-1}}}

{{{9(x-3)^2 +16 (y - 2)^2 =144}}}..........both sides divide by {{{144}}}

{{{9(x-3)^2/144 +16 (y - 2)^2/144 =144/144}}}

{{{(x-3)^2/16 + (y - 2)^2/9 =1}}}

=> {{{h=3}}}, {{{k=2}}}, 
{{{a=sqrt(16)}}}=>{{{a=4}}}
{{{b=sqrt(9)}}}=>{{{b=3}}}

{{{a>b}}}therefore {{{a}}} is semi-major axis and {{{b}}} is semi-minor axis 

{{{c=sqrt(a^2-b^2)}}}=>{{{c=sqrt(4^2-3^2)}}}=>{{{c=sqrt(7)}}}

1. Center: ({{{3}}},{{{2}}})

2. Foci:  ({{{h+c}}}, {{{k}}} ), ({{{h-c}}}, {{{k}}} )

F1 ({{{3+sqrt(7)}}}, {{{2}}} ), ({{{3-sqrt(7)}}}, {{{2}}} )
F2 ({{{5.7}}}, {{{2}}} ), ({{{-0.4}}}, {{{2}}} )



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

V1 ({{{3+4}}},{{{2}}}) => ({{{7}}},{{{2}}})
V2 ({{{3-4}}},{{{2}}}) => ({{{-1}}},{{{2}}})

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



cV1 ({{{3}}},{{{2+3}}}) =>({{{3}}},{{{5}}})
cV2({{{3}}},{{{2-3}}})=>({{{3}}},{{{-1}}})

5. Endpoints of Latus Rectum

 foci are  ({{{5.7}}}, {{{2}}} ),  ({{{-0.4}}}, {{{2}}} )

endpoints will lie on intersection of the line {{{x=5.7}}} and the line {{{x=-0.4}}} with ellipse

{{{(x-3)^2/16 + (y - 2)^2/9 =1}}}...substitute {{{x}}}
{{{(5.7-3)^2/16 + (y - 2)^2/9 =1}}}...solve for {{{y}}}
{{{0.455625+ (y - 2)^2/9 =1}}}
{{{ (y - 2)^2 =9(1-0.455625)}}}
{{{ (y - 2)^2 =4.899375}}}
{{{ y  =sqrt(4.899375)+2}}}
{{{ y  =4.2}}} or {{{ y=-0.2}}} 

{{{(x-3)^2/16 + (y - 2)^2/9 =1}}}...substitute {{{x}}}
{{{(-0.4-3)^2/16 + (y - 2)^2/9 =1}}}...solve for {{{y}}}
{{{0.7225+ (y - 2)^2/9 =1}}}
{{{ (y - 2)^2 =9(1-0.7225)}}}
{{{ (y - 2)^2 =2.4975}}}
{{{ y  =sqrt(2.4975)+2}}}
{{{ y  =3.6}}} or {{{ y  =0.4}}} 


E1 ({{{5.7}}}, {{{4.2}}} )
E2 ({{{5.7}}}, {{{-0.2}}} )
E3  ({{{-0.4}}}, {{{3.6}}} )
E4 ({{{-0.4}}}, {{{0.4}}} )


6. Directrices:
{{{x=3-16sqrt(7)/7}}}=>approximately {{{x=-3.1}}}
{{{x=3+16sqrt(7)/7}}}=>approximately {{{x=9.1}}}


7. Eccentricity: {{{sqrt(7)/4}}} approximately {{{0.7}}}

8. Length of LR:{{{9/2}}}

9. Length of Major Axis:{{{4}}}

10.Length of Minor Axis:{{{3}}}


{{{drawing ( 600, 600, -10, 10, -10, 10,
circle(3, 2,.13),locate(3, 2,C),
circle(5.7, 4.2,.13),locate(5.7, 4.2,E1),
circle(5.7, -0.2,.13),locate(5.7, -0.4,E2),
circle(-0.4, 3.6,.13),locate(-1, 3.9,E3),
circle(-0.4, 0.4,.13),locate(-1, 0.4,E4),
circle(5.7, 2,.13),locate(5.7, 2,F),
circle(-0.4, 2,.13),locate(-0.4, 2,F),

circle(7, 2,.13),locate(7, 2,V),
circle(-1, 2,.13),locate(-1, 2,V),
circle(3, 5,.13),locate(3,5,cV),
circle(3, -1,.13),locate(3,-1,cV),

graph( 600, 600, -10, 10, -10, 10,-sqrt( 9(1-(x-3)^2/16))+2,sqrt( 9(1-(x-3)^2/16))+2 )) }}}