Final answer:
The task is to arrange equations of ellipses by increasing eccentricity, a property indicating the deviation from a circular shape. The calculation involves comparing the formula e = sqrt(1 - (b^2/a^2)) for each provided ellipse.
Step-by-step explanation:
The mathematical concept of eccentricity in ellipses, which is a measure of how much an ellipse deviates from being circular. Eccentricity is denoted by the letter 'e' and ranges from 0 for a perfect circle to nearly 1 for a very elongated ellipse. To arrange the equations of ellipses in order of increasing eccentricities, one would compare the values of 'e' for each given ellipse.
Recall that for an ellipse with a major axis length of 'a' and a minor axis length of 'b', the eccentricity can be calculated using the formula e = √{1 - (b^2/a^2)}. An ellipse with a larger 'a' relative to 'b' will have a greater eccentricity. Therefore, to organize the equations by eccentricity, you need to either calculate 'e' from each equation if 'a' and 'b' are given, or compare their ratio if the equation of the ellipse is provided in standard form.