Did you know?
Did you know that the foci, vertices, transverse axis, conjugate axis, and center can be determined by analyzing the equation of an ellipse? For example, in the equation (x-2)²/4 - (y-4)²/216 = 1, we can identify that the center of the ellipse is located at the coordinates (2, 4).
The major axis, commonly known as the transverse axis, is oriented vertically in this case. It measures 2 times the square root of 216, equal to approximately 29.39 units. The vertices, which lie on the transverse axis, can be found by adding and subtracting this value to/from the y-coordinate of the center. In this case, the vertices would be located at (2, 4±29.39).
The minor axis, or conjugate axis, is horizontally oriented in this equation. Its length is 2 times the square root of 4, which equals 4 units. Similarly, the conjugate axis intersects the center and its length extends in both directions. Thus, the conjugate axis would span between (2±4, 4).
Additionally, the foci of the ellipse can be determined. The formula to calculate the distance from the center to the foci is given by c = √(a² - b²), where a represents the length of the transverse axis and b represents the length of the conjugate axis. In this case, c equals approximately 28.95 units. By using this value, we can find the foci along the same transverse axis as the vertices, with coordinates (2, 4±28.95).
By understanding the different components of the ellipse equation and how they relate to its shape, size, and positioning, we can unravel more information about this geometric figure.
