Question 1197350

the equation (hyperbola)  is:

{{{(x-h)^2/a^2 -(y-k)^2/b^2=1}}}


Find the equation (hyperbola) if the

asymptotes {{{y - 3 = (sqrt(13)/6) (x-2) }}} => {{{y  = (sqrt(13)/6) (x-2) +3}}}

and focus at ({{{9}}}, {{{3}}})


The first asymptote is:

{{{y=  - (b/a )(x-h)+k}}}

compare to {{{y  = (sqrt(13)/6) (x-2) +3 }}}and you see that

{{{k=3}}}

 {{{-(b/a )= (sqrt(13)/6)}}}  =>{{{a=6}}} and{{{b= sqrt(13)}}} 

take into account different properties of a hyperbola:

{{{(h-9)^2=a^2+b^2}}}

{{{(h-9)^2=49}}}

{{{(h-9)=7}}}  or{{{ (h-9)=-7}}}

if {{{(h-9)=7}}}=>{{{h=16}}}

if {{{(h-9)=-7}}} => {{{h=2 }}}


substitute in equation: {{{h=2}}}, {{{k=3}}}, {{{a=6}}} and {{{b=sqrt(13)}}}

{{{(x-2)^2/6^2-(y-3)^2/(sqrt(13))^2=1}}}

{{{(x-2)^2/36-(y-3)^2/13=1}}}


answer: {{{(x-2)^2/36-(y-3)^2/13=1}}}