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

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Question 1197350: Find the equation (hyperbola)
asymptotes y - 3 = √13/6 (x-2) and focus at (9, 3)
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

the equation (hyperbola) is:
%28x-h%29%5E2%2Fa%5E2+-%28y-k%29%5E2%2Fb%5E2=1

Find the equation (hyperbola) if the
asymptotes y+-+3+=+%28sqrt%2813%29%2F6%29+%28x-2%29+ => y++=+%28sqrt%2813%29%2F6%29+%28x-2%29+%2B3
and focus at (9, 3)

The first asymptote is:
y=++-+%28b%2Fa+%29%28x-h%29%2Bk
compare to y++=+%28sqrt%2813%29%2F6%29+%28x-2%29+%2B3+and you see that
k=3
-%28b%2Fa+%29=+%28sqrt%2813%29%2F6%29 =>a=6 andb=+sqrt%2813%29
take into account different properties of a hyperbola:
%28h-9%29%5E2=a%5E2%2Bb%5E2
%28h-9%29%5E2=49
%28h-9%29=7 or+%28h-9%29=-7
if %28h-9%29=7=>h=16
if %28h-9%29=-7 => h=2+

substitute in equation: h=2, k=3, a=6 and b=sqrt%2813%29
%28x-2%29%5E2%2F6%5E2-%28y-3%29%5E2%2F%28sqrt%2813%29%29%5E2=1
%28x-2%29%5E2%2F36-%28y-3%29%5E2%2F13=1

answer: %28x-2%29%5E2%2F36-%28y-3%29%5E2%2F13=1