Question 1197351
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If you use the "√" symbol for a square root, then use parentheses to make it clear exactly what part of the expression is under the radical.  "√13/6" could mean either sqrt(13)/6 or sqrt(13/6).<br>
A coefficient like sqrt(13/6) would probably not be given in a problem, since it is not in simplified form; so I will assume the coefficient is sqrt(13)/6.<br>
You show the equation of one of the asymptotes; I further assume that the asymptotes (plural) have equations with coefficients sqrt(13)/6 and -sqrt(13)/6.<br>
The equations of the asymptotes tell us the center of the hyperbola is (2,3); and the focus at (9,3) tells us the branches of the hyperbola open right and left.  So we know the equation of the hyperbola has the form<br>
{{{(x-2)^2/a^2-(y-3)^2/b^2=1}}}<br>
The slopes of the asymptotes tell us that {{{b/a=sqrt(13)/6}}}.<br>
Finally, with the center at (2,3) and one focus at (9,3), we know that c, the distance from the center to each focus, is 7.<br>
So we have...
b/a = sqrt(13)/6;
c = 7;
and, for a hyperbola, {{{c^2 = a^2+b^2}}}<br>
Then the simple observation that {{{7^2 = sqrt(13)^2+6^2}}} tells us that b = sqrt(13) and a = 6.<br>
So the equation of the hyperbola is<br>
{{{(x-2)^2/36-(y-3)^2/13=1}}}<br>