F1F2 is the left and right focus of the hyperbola, P is a point on the hyperbola, angle f1pf2 = 60 degrees, triangle pf1f2 = 12 √ 3, and eccentricity is 2
1/2PF1×PF2×sin60=12√3
PF1×PF2=48
c/a=2
c=2a
|PF1-PF2|=2a
PF1²-2PF1×PF2+PF2²=c²
PF1²+PF2²=c²+96
According to the cosine theorem
cos60=(PF1²+PF2²-F1F2²)/(2PF1×PF2)
1/2=(c²+96-4c²)/96
48=96-3c²
3c²=48
c²=16
c=4
a=c/2=2
b²=c²-a²=16-2²=12
Hyperbolic equation: X & sup2 / 4-y & sup2 / 12 = 1
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