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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.

**00:00:00**In this section, the speaker is discussing the equation of a vertical hyperbola that is centered at the origin. To deduce the equation, the speaker considers the fixed points, such as the foci, which are located at a distance "c" from the origin, with coordinates (-c, 0) and (0, c), respectively. The distance between these two fixed points is 2, so they are located at coordinates (-c, 0) and (0, c), where c is the distance between the foci. The speaker also addresses the relationship between the fixed points, the vertices, and the center of the hyperbola. The two vertices are located at (-c, 0) and (0, c), while the center is at the origin. The speaker notes that the distance between the center and the vertices is the same as the distance between the foci. To deduce the equation, the speaker starts by noting that the distance between any point on the hyperbola and the foci is constant. This distance can be expressed as the difference between the coordinates of the foci and the point of interest, multiplied by 2. Starting with this relationship, the speaker considers the distance between a point (x, y) on the hyperbola and the foci (0, -c) and (0, c). By simplifying this expression, the speaker arrives at the equation: (x - 0)^2 + (y - 0)^2 = 2a^2. The speaker then notes that the distance between the two foci can be expressed as the square root of (c^2 + c^2) = 2c, and the distance between the point and the center of the hyperbola can be expressed as the square root of (x - 0)^2 + (y - 0)^2 = x^2 + y^2. Finally, by combining these equations and simplifying, the speaker arrives at the standard form of the equation of a vertical hyperbola centered at the origin, which is x^2/(x^2 - 2c) + y^2 = 1.**00:05:00**This video provides a clear explanation of how to solve for the roots of a quadratic equation. It starts with the formula that the first equation is equal to the first squared plus twice the first squared times the second squared. The video then provides a step-by-step process for solving for the roots by canceling the second equation, taking the second equation to the square, and finally factoring out the terms that do not have roots. The final result is a simplified expression for one of the roots, which is defined by the choices made in the solution process. Overall, the video is a helpful resource for anyone looking to understand how to solve quadratics step-by-step.**00:10:00**This section of a video on parabolas outlines how to solve a vertical parabola in standard form. By first introducing a common factor into the equation, the terms can be simplified. This is followed by dividing both sides of the equation by the common factor and simplifying further into the standard form. It is noted that the resulting variables are positive and square of a common term. Overall, the process is similar to that of solving a horizontal parabola, with the only difference being the change in direction for the roots of the variable.**00:15:00**In this section, the speaker discusses the equation of a vertical hyperbola centered at the origin. The hyperbola's equation is y = b/x^2, where b is the constant. For large x values, it becomes y = 0 or y = ∞. However, for large positive or negative x values, it becomes y = b. If the equation is multiplied by a constant, it becomes y = a + b/x^2 or y = a - b/x^2, where a is another constant.

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