The standard form of the equation of an ellipse centered at the origin (0,0) depends on whether the major axis is horizontal or vertical.
In our case, we have vertices at (6, 0) and (0, 4). The distance from the center to the vertices on the x-axis is 6, which indicates that the major axis is horizontal. The distance from the center to the vertices on the y-axis is 4, indicating the minor axis. This gives us:
- Length of the semi-major axis, a = 6
- Length of the semi-minor axis, b = 4
In the standard form, the equation of an ellipse is expressed as:
(x^2 / a^2) + (y^2 / b^2) = 1
Substituting in our values for a and b:
- a^2 = 6^2 = 36
- b^2 = 4^2 = 16
Now, replacing a and b in the standard form equation:
(x^2 / 36) + (y^2 / 16) = 1
Thus, the equation of the ellipse in standard form is:
(x^2 / 36) + (y^2 / 16) = 1
This equation captures the essence of the ellipse, showing that it is centered at the origin with its major axis along the x-axis and its minor axis along the y-axis.