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The video titled "Ecuación de la hipérbola vertical centrada en el origen" focuses on the equation of a vertical hyperbola centered at the origin. The speaker starts by discussing the fixed points, vertices, and center of the hyperbola. The fixed points are located at (-c, 0) and (0, c), while the center is at the origin. The two vertices are located at (-c, 0) and (0, c), and the distance between the center and the vertices is the same as the distance between the foci. The speaker then moves on to deducing the equation of the hyperbola. The distance between any point on the hyperbola and the foci is constant, which can be expressed as the difference between the coordinates of the foci and the point of interest, multiplied by 2. This expression leads the speaker to arrive at the equation: (x - 0)^2 + (y - 0)^2 = 2a^2. The speaker then notes that this equation can be simplified to x^2 + y^2 = 2a^2, which is the standard form of the equation of a vertical hyperbola centered at the origin.
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