Question 1089559
find the equation for the hyperbola :{{{(x-h)^2/a^2-(y-k)^2/b^2=1}}}

if a center at ( {{{1}}},{{{1}}}) =({{{h}}},{{{k}}}), we have

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


if an {{{x}}}- intercept at ({{{2}}},{{{0}}}), we have 

{{{(2-1)^2/a^2-(0-1)^2/b^2=1}}}

{{{1/a^2-1/b^2=1}}}.....eq.1


and if one vertex at ({{{5/4}}}, {{{1}}}), we have 


{{{(5/4-1)^2/a^2-(1-1)^2/b^2=1}}} ...eq.2 ...solve or {{{a}}}

{{{(5/4-4/4)^2/a^2-0/b^2=1}}}

{{{(1/4)^2/a^2=1}}}

{{{a^2=(1/16)}}}


go to

{{{1/a^2-1/b^2=1}}}.....eq.1 plug in {{{a^2=1/16}}} and solve or {{{b}}}

{{{1/16-1/b^2=1}}}
{{{16-1/b^2=1}}}
{{{16-1=1/b^2}}}
{{{1/b^2=15}}}
{{{b^2=1/15}}}

your equation is:

{{{(x-1)^2/(1/16)-(y-1)^2/(1/15)=1}}}

{{{16(x-1)^2-15(y-1)^2=1}}}


{{{drawing( 600, 600, -5, 5, -5, 5,
circle(1,1,.012), locate(1,1,C(1,1)),
circle(5/4,1,.012), locate(1.3,1,V(5/4,1)),
circle(2,0,.012), locate(2,-0.3,p(2,0)),
 graph( 600, 600, -5, 5, -5, 5, (1/15) (sqrt(15) sqrt(16x^2 - 32x + 15) + 15), (1/15) (15 - sqrt(15) sqrt(16x^2 -32x + 15)))) }}}