what happens when the major and minor axies are exactly the same length?
When the major and minor axes of an ellipse are exactly the same length, the ellipse becomes a special case known as a circle. In a circle, every point on the boundary is equidistant from the center.
To understand why a circle forms when the major and minor axes are equal, let's first define the terms. The major axis is the longest diameter of the ellipse, which passes through the center and divides it into two equal halves. The minor axis is the shortest diameter of the ellipse, perpendicular to the major axis, also dividing the ellipse into two equal halves.
When the major and minor axes have the same length, they are essentially identical. This means that the distance from any point on the boundary of the ellipse to the center is the same in all directions. As a result, the ellipse transforms into a circle, where the radius is equal at every point.
To verify this mathematically, you can use the equation of an ellipse, which is:
(x^2 / a^2) + (y^2 / b^2) = 1
Here, 'a' represents the radius along the major axis and 'b' represents the radius along the minor axis. When a = b, the equation simplifies to:
(x^2 / a^2) + (y^2 / a^2) = 1
This equation of a circle, where 'a' represents the common radius.