answer:Since the absolute value of the difference in distances of point P to F₁ (-5, 0) and to F₂ (5, 0) satisfies |PF₁| - |PF₂| = 6, and the distance between F₁ and F₂ is 10 which is greater than 6, the trajectory of point P is the right branch of the hyperbola with foci at F₁ (-5, 0) and F₂ (5, 0) and a real-axis length of 6. Using the definition of a hyperbola and the given geometric condition for point P, we can determine that the trajectory is part of a hyperbola. The standard equation of the hyperbola is thus given by: frac {x^{2}}{9}-frac {y^{2}}{16}=1 (x geq 3) Therefore, the correct option is D: boxed{D: frac {x^{2}}{9}-frac {y^{2}}{16}=1 (x geq 3)} This problem tests the understanding of the definition and standard equation of a hyperbola, as well as the method to find the equation of the trajectory of a moving point defined by a hyperbola.

answer:Given: angle A=60^{circ}, side AB=2, S_{triangle ABC}= frac{ sqrt {3}}{2}, We know that the area of a triangle is given by S_{triangle ABC}= frac{1}{2}AB cdot AC cdot sin A. Substituting the given values, we get frac{ sqrt {3}}{2}= frac{1}{2} cdot 2 cdot AC cdot frac{ sqrt {3}}{2}. Solving for AC, we find AC=1. Next, we use the cosine rule to find BC. The cosine rule states that BC^{2}=AB^{2}+AC^{2}-2AB cdot AC cdot cos A. Substituting the known values, we get BC^{2}=2^{2}+1^{2}-2 cdot 2 cdot 1 cdot cos 60^{circ} = 4 + 1 - 2 = 3. Taking the square root of both sides, we find BC = sqrt {3}. Therefore, the length of side BC is boxed{sqrt {3}}. To solve this problem, we first used the formula for the area of a triangle to find the length of side AC. Then, using the cosine rule and the lengths of sides AB and AC, we were able to find the length of side BC. This problem tests our understanding of the cosine rule, the formula for the area of a triangle, and the values of trigonometric functions for special angles. Proficiency in using the cosine rule is key to solving this problem.

answer:We have to find perfect-square integers close to 350, and also ensure that these integers are multiples of 4. 1. Calculate square roots around 350: [ sqrt{350} approx 18.7 ] So, we consider the integers 18 and 19 for calculation. 2. Calculate squares of these numbers: [ 18^2 = 324, quad 19^2 = 361 ] 3. Check if these are multiples of 4: - 324 div 4 = 81, which is an integer, so 324 is a multiple of 4. - 361 div 4 = 90.25, which is not an integer, so 361 is not a multiple of 4. 4. Determine the distance of 324 and 361 from 350: - |350 - 324| = 26 - |350 - 361| = 11 Since 361 is closer to 350 but not a multiple of 4, we need the closest perfect square that is a multiple of 4. This is 324. Conclusion: [ boxed{324} ]

answer:To find the speed of the train, we can use the formula: Speed = Distance / Time We know the distance (length of the train) is 250 meters, and the time taken to cross the pole is 10 seconds. First, we convert the distance from meters to kilometers by dividing by 1000 (since 1 km = 1000 m): Distance in km = 250 meters / 1000 = 0.25 km Next, we convert the time from seconds to hours by dividing by 3600 (since 1 hour = 3600 seconds): Time in hours = 10 seconds / 3600 = 1/360 hours Now we can calculate the speed: Speed = 0.25 km / (1/360) hours Speed = 0.25 km * (360/1) hours Speed = 90 km/hr Therefore, the speed of the train is boxed{90} km/hr.