Is the ellipse more circular when the tacks are closer together or farther apart?
If you did the experiment, you should know. Thinking about it what happens when you move the thumbtacks together will also help.
To determine whether an ellipse is more circular when the tacks are closer together or farther apart, we need to understand how the position of the tacks affects the shape of the ellipse. The tacks are used to help draw the ellipse, with a string looped around them.
An ellipse is a curve that is symmetric about its center point, with two focal points located inside the ellipse. The distance between these two focal points is a constant and is known as the major axis. When the tacks are closer together, the major axis becomes shorter, while a greater distance between the tacks results in a longer major axis.
The ratio of the length of the minor axis to the major axis determines the eccentricity of an ellipse. The eccentricity of a circle is zero, meaning the minor axis is equal to the major axis. As the eccentricity of an ellipse approaches zero, it becomes closer to a circle.
Therefore, when the tacks are closer together, the major axis becomes shorter, increasing the likelihood that the ellipse will approach a circular shape. Conversely, when tacks are farther apart, the major axis becomes longer, resulting in a less circular ellipse.
So, to answer your question, the ellipse is more circular when the tacks are closer together.