Chapter 9.2, Problem 68E

To determine

To order: the eccentricities of the ellipse from least to greatest.

Explanation of Solution

Given information:

Given figure

  

Calculation:

The eccentricity $e$ of an ellipse is defined by the ratio $e=\frac{c}{a}$ ; and its value is always less than 1, that is $0<e<1$ .

In a ellipse the foci are located along the major axis between center and vertices. Hence

  $0<c<a$

Thus the eccentricity of an ellipse depends on the valve of $c$ and the value of $a$ . For an ellipse which is nearly circular the foci are located very near to the centre; so the value of $c$ is very small in comparison to the value of $a$ . This implies that the eccentricity will be close to 0. In case where ellipse is too flat, the foci are located close to vertices implying that the value of $c$ is close the value of $a$ . Hence the eccentricity will be close to 1.

Following the explanation given above the eccentricity of ellipse $B$ will be least and that of ellipse $C$ will be greatest. Therefore the order of eccentricities of the given ellipse from least to greatest is

Ellipse B<Ellipse A<Ellipse C