Question 1043408
 Find the standard equation,center,foci,asymptotes of 

{{{3x^2-2y^2-42x-10y=-64 }}}...group alike terms

{{{(3x^2-42x)-(2y^2+10y)=-64}}}..........complete squares 

{{{3(x^2-14x+_)-2(y^2+5y+_)=-64 }}}

{{{3(x^2-14x+7^2)-3*7^2-2(y+5/2)^2-2*(25/4)=-64 }}}

{{{3(x-7)^2-3*49-2(y+5/2)^2-(25/2)=-64 }}}

{{{3(x-7)^2-147-2(y+5/2)^2-(25/2)=-64}}} 

{{{3(x-7)^2-2(y+5/2)^2=-64 +147+25/2}}}......both sides multiply by {{{2}}}

{{{6(x-7)^2-4(y+5/2)^2=-128 +296+25}}}

{{{6(x-7)^2-4(y+5/2)^2=193}}}.....both sides divide by {{{193}}}

{{{6(x-7)^2/193-4(y+5/2)^2/193=193/193}}}

{{{(6/193)(x-7)^2-(4/193)(y+5/2)^2=1 }}}

{{{(x-7)^2/(193/6)-(y+5/2)^2/(193/4)=1 }}}->hyperbola, the standard equation

{{{h=7}}}
{{{k= -5/2}}}
{{{a = sqrt(193/6)}}} and {{{b =sqrt(193/4)}}}

{{{c^2 = a^2 + b^2}}}

{{{c^2 =(sqrt (193/6))^2 + (sqrt(193/4))^2}}}

{{{c^2 =193/6 + 193/4}}}

{{{c^2 =965/12}}}

{{{c=sqrt(965/12)}}}

so, the center is at ({{{h}}},{{{k}}})= ({{{7}}},{{{-5/2}}})

foci: since {{{c = sqrt(965/12)}}} ,  the foci will be {{{sqrt(965/12)}}}  units to either side of the center, and it must be at 

({{{7-sqrt(965/12)}}}, {{{-5/2}}}) and  ({{{7+sqrt(965/12)}}}, {{{-5/2}}})
or ({{{-1.97}}},{{{-2.5}}}) and ({{{15.96}}}, {{{-2.5}}})

 Since the vertices are {{{a =sqrt( 193/6)}}} units to either side of the center ({{{h}}},{{{k}}})= ({{{7}}},{{{-5/2}}}), then they are at
 ({{{7-sqrt(193/6)}}}, {{{-5/2}}}) and at ({{{7+sqrt(193/6)}}}, {{{-5/2}}})
or ({{{1.33}}}, {{{-2.5}}}) and at ({{{12.67}}}, {{{-2.5}}})

Since the {{{a^2}}} went with the {{{x}}} part of the equation, then {{{a=sqrt(193/6)}}} is in the denominator of the slopes of the asymptotes, giving me {{{m =b/a= sqrt(193/4)/ sqrt(193/6)=sqrt(6/4)=sqrt(3/2)}}}; so, {{{m=sqrt(3/2)}}}or {{{m=-sqrt(3/2)}}}. Keeping in mind that the asymptotes go through the center of the hyperbola, the asymptes are then given by the straight-line equations 
 
The asymptotes are {{{y=(b/a)(x-h)+k}}}, and they are:

{{{y= sqrt( 3/2)(x -7 ) -5/2 }}}
and 
{{{y=  -sqrt(3/2)(x -7 ) -5/2 }}}



{{{ graph( 600, 600, -30, 30, -30,30, sqrt(((x-7)^2/(193/6)-1 )(193/4))-5/2, -sqrt(((x-7)^2/(193/6)-1 )(193/4))-5/2,  sqrt(3/2)(x -7 )-5/2 ,-  sqrt(3/2)(x -7 ) -5/2 ) }}}