Final answer:
The question has a typo and doesn't present a conventional ellipse equation. Still, general properties of an ellipse are discussed, such as the center located at the origin, the eccentricity, the latus rectum, and associated eccentric angles.
Step-by-step explanation:
The question is asking to find the coordinates of the center of an ellipse and the eccentric angles of the extremities of the latus rectum of the ellipse represented by the equation x² - 2cos²β = 1.
However, there seems to be a typo in the question, as the equation does not describe an ellipse in its conventional form which should be (x²/a²) + (y²/b²) = 1. Nevertheless, we can still discuss some general properties of ellipses related to the question.
The center of an ellipse is the midpoint between its foci and, in a standard ellipse equation, it is at the origin (0,0). The latus rectum of an ellipse is a line segment perpendicular to the major axis through a focus, and its extremities are on the ellipse. To find these points, you would normally set y equal to the distance from the center to a focus (c) and solve for x, using the ellipse equation in the standard form.
The notion of eccentricity is important for understanding ellipses. It is denoted by e and is calculated as e = f/a, where f is the distance from the center to a focus, and a is half the length of the major axis. The eccentricity determines the shape of the ellipse; if e = 0, the ellipse is a circle. The eccentric angles are angles formed by lines drawn from the center of the ellipse to the endpoints of the latus rectum.