The left and right focus of the hyperbola x2a2 − y2b2 = 1 is F1, F2, P is a point on the hyperbola, satisfying | PF2 | = | F1F2 |, and the straight line Pf1 is tangent to the circle x2 + y2 = A2, then the eccentricity e of the hyperbola is () A. 3B. 233C. 53D. 54 | Math enthusiast | Mathematical art

The left and right focus of the hyperbola x2a2 − y2b2 = 1 is F1, F2, P is a point on the hyperbola, satisfying | PF2 | = | F1F2 |, and the straight line Pf1 is tangent to the circle x2 + y2 = A2, then the eccentricity e of the hyperbola is () A. 3B. 233C. 53D. 54

Let Pf1 be tangent to the circle at point m and f2h be perpendicular to Pf1 through F2. Then h is the middle point of Pf1, and ∵ PF2 | = | F1F2 |, ∵ pf1f2 is an isosceles triangle, and | f1m | & nbsp; = 14 | & nbsp; Pf1 |, ∵ right triangle f1mo, in which | f1m | 2 = c2-a2, | f1m | = b = 14 | Pf1 |, | 2A = 4b-2c ∵ C2 = A2 + B2

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1. It is known that F1 and F2 are the two focuses of the hyperbola x ^ 2 / 16 - y ^ 2 / 9 = 1, and P is a point on the hyperbola, It is known that F1 and F2 are two focal points of hyperbola x ^ 2 / 16 - y ^ 2 / 9 = 1, P is a point on hyperbola, and Pf1 ⊥ PF2. Find the area of △ pf1f2
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