SOLUTION: Find the equation (hyperbola) if the asymptotes y - 3 = √13/6 (x-2) and focus at (9, 3)

Question 1197351: Find the equation (hyperbola) if the
asymptotes y - 3 = √13/6 (x-2) and focus at (9, 3)
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

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).

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.

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.

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

The slopes of the asymptotes tell us that .

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.

So we have...
b/a = sqrt(13)/6;
c = 7;
and, for a hyperbola,

Then the simple observation that tells us that b = sqrt(13) and a = 6.

So the equation of the hyperbola is

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