Question 1181019
Given the equation 

{{{36x^2-36y^2=1296}}}} 

Sketch and determine the parts of the hyperbola.

{{{36x^2-36y^2=1296}}}} .......write in standard form {{{(x-h)^2/a^2-(y-k)^2/b^2=1}}}, both sides divide by {{{1296}}}         

{{{36x^2/1296-36y^2/1296=1296/1296}}}}

{{{x^2/36-y^2/36=1}}}}

=>{{{h=0}}} and {{{k=0}}}
=>{{{a^2=36}}} => {{{a=6}}}
=>{{{b^2=36}}} => {{{b=6}}}

{{{c=sqrt(36+36)}}}=>{{{c=sqrt(2*36)}}}=>{{{c=6sqrt(2)}}}

Parts of the Hyperbola

1. Center: ({{{0}}},{{{0}}})

2. Foci: is {{{c, k)}}} or {{{-c, k}}} 
 
({{{6sqrt(2)}}},{{{0}}}) and ({{{-6sqrt(2)}}},{{{0}}})

3. Vertices: one vertex is at ({{{a}}}, {{{0}}}), and the other is at ({{{-a}}}, {{{0}}})

since {{{a=6}}}
 Vertices:({{{6}}}, {{{0}}}) and ({{{-6}}}, {{{0}}})
Co-vertices: ({{{0}}},{{{-6}}}), ({{{0}}},{{{6)

4. End Points of Conjugate Axis 
B1= ({{{0}}},{{{-6}}})
B2=({{{0}}},{{{6)

5. End Points of Latus Rectum E1, E2, E3, E4
find intercepts of the line {{{x=6sqrt(2)}}} and {{{x=-6sqrt(2)}}} that passes through foci and {{{x^2/36-y^2/36=1}}}}

{{{(6sqrt(2))^2/36-y^2/36=1}}}}
{{{(36*2)/36-y^2/36=1}}}}
{{{2-y^2/36=1}}}}
{{{2-1=y^2/36}}}}

{{{1=y^2/36}}}}
{{{36=y^2}}}}

{{{y=6}}} or {{{y=-6}}}

points are:
E1=({{{6sqrt(2)}}}, {{{6}}})
E2=({{{6sqrt(2)}}}, {{{-6}}})
E3=({{{-6sqrt(2)}}}, {{{6}}})
E4=({{{-6sqrt(2)}}}, {{{-6}}})

6. Asymptotes:
 {{{y = (b/a)x}}}=>{{{y = (6/6)x}}}=>{{{y = x}}}
    {{{y = -(b/a)x}}}=>{{{y =- (6/6)x}}}=>{{{y = -x}}}

7. Eccentricity:
{{{e = sqrt(a^2+b^2)/a}}}
{{{e = sqrt(6^2+6^2)/6}}}
{{{e = 6sqrt(2)/6}}}
{{{e=sqrt(2)}}}≈{{{1.41421}}}

8. Length of LR:
The length of the Latus Rectum is {{{2b^2/a= (26^2)/6=2*6=12}}}
9. Length of Conjugate Axis: {{{12}}}

10. Length of Traverse Axis:{{{12}}}




   {{{drawing ( 600, 600, -15, 15, -15, 15,
circle(6sqrt(2),0,.13),locate(6sqrt(2),0.9,F(6sqrt(2),0)),

circle(-6sqrt(2),0,.13),locate(-6sqrt(2),0.9,F(-6sqrt(2),0)),

circle(6,0,.13),locate(6,0,v(6,0)),
circle(-6,0,.13),locate(-6,0,v(-6,0)),


circle(0,6,.13),locate(0.2,6,B1),
circle(0,-6,.13),locate(0.2,-6,B2),

circle(6sqrt(2),6,.13),locate(6sqrt(2),6,E1),
circle(6sqrt(2),-6,.13),locate(6sqrt(2),-6,E2),

circle(-6sqrt(2),6,.13),locate(-6sqrt(2),6,E3),
circle(-6sqrt(2),6,.13),locate(-6sqrt(2),-6,E4),

blue(line(-6sqrt(2),6,-6sqrt(2),-6)), blue(line(6sqrt(2),6,6sqrt(2),-6)),
graph( 600, 600, -15, 15, -15, 15,- sqrt(x^2-36),sqrt(x^2-36) )) }}}