Question 1010718

{{{5x^2-5y^2+40x-20y+35=0}}}.........all terms divide by {{{5}}}

{{{x^2+8x-y^2-4y+7=0}}}

{{{(x^2+8x)-(y^2+4y)+7=0}}}...complete squares

{{{(x^2+8x+b^2)-b^2-(y^2+4y+b^2)-b^2+7=0}}}

{{{(x^2+8x+4^2)-4^2-(y^2+4y+2^2)-2^2+7=0}}}

{{{(x+4)^2-16-(y+2)^2-4+7=0}}}

{{{(x+4)^2-(y+2)^2-20+7=0}}}

{{{(x+4)^2-(y+2)^2-13=0}}}

{{{(x+4)^2-(y+2)^2=13}}}=> both sides divide by {{{13}}}

{{{(x+4)^2/13-(y+2)^2/13=1}}}=> so, we have a hyperbola with {{{h=-4}}},{{{k=-2}}}

the center is at ({{{-4}}},{{{-2}}}) 

semimajor axis length |{{{a= sqrt(5)}}}~~{{{a=2.24}}}
semiminor axis length | {{{b=sqrt(5)}}}~~{{{b=2.24}}}


foci is fixed distance {{{c}}} from the center 
{{{c^2 = b^2 + a^2}}}=>{{{c^2 = (sqrt(5))^2 + (sqrt(5))^2}}}=>{{{c^2 = 5 + 5}}}
{{{c=sqrt(10)}}}

so, foci is at:
({{{-4-sqrt(10)}}},{{{ -2}}})  and  ({{{-4+sqrt(10)}}},{{{ -2}}})~~({{{-7.2}}}, {{{-2}}})  and  ({{{-0.8}}}, {{{-2}}})

vertices | ({{{-4-sqrt(5)}}},{{{ -2}}})  and  ({{{-4+sqrt(5)}}}, {{{-2}}})~~({{{-6.2}}}, {{{-2}}}) and  ({{{-1.8}}}, {{{-2}}})


{{{ graph( 600, 600, -15, 10, -10, 10, sqrt(((x+4)^2/13-1)13)-2,-sqrt(((x+4)^2/13-1)13)-2 ) }}}