Chapter 14.8, Problem 19WE

a.

To determine

To describe the symmetries of ellipse shown.

Answer to Problem 19WE

The ellipse has two symmetries one with horizontal line passing through the centre of ellipse and other with the vertical line passing through the centre of ellipse.

Explanation of Solution

Given information:

The given statement is that describe symmetries of the ellipse shown.

The ellipse has two line symmetries one with horizontal line passing through the centre of ellipse and other with the vertical line passing through the centre of ellipse as shown in figure below. And the ellipse also has a point symmetry at the intersection of its major and minor axis.

  

b.

To determine

To interpret the volume formula of ellipse when $a=b$ .

Answer to Problem 19WE

The formula obtained when $a=b$ is formulae of volume of a sphere with radius equals to a . hence, when major axis of ellipsoid becomes equal to its minor axis it the ellipsoid becomes a sphere.

Explanation of Solution

Given information:

The given statement is if the ellipse is rotated in space about one of its symmetry lines anellipsoid is formed. Its volume is $V=\frac{4}{3}\pi a^{2}b$ .

To interpret the formulae substitute $a=b$ in $V=\frac{4}{3}\pi a^{2}b$

  $\begin{array}{l}V=\frac{4}{3}\pi a^{2}b\\ V=\frac{4}{3}\pi a^2\cdot a\left[a=b\right]\\ V=\frac{4}{3}\pi a^3\end{array}$

The formula obtained when $a=b$ is formulae of volume of a sphere with radius equals to a . hence, when major axis of ellipsoid becomes equal to its minor axis it the ellipsoid becomes a sphere.

c.

To determine

To describe the symmetries of ellipsoid.

Answer to Problem 19WE

The ellipsoid have three line symmetries with respect to three mutually perpendicular axis which intersect each other at the centre of ellipsoid.

Explanation of Solution

Given information:

The given statement is that describe symmetries of the ellipsoid.

  

The ellipsoid has many number of plane symmetry about the many planes that contain the b and plane symmetry about the single plane that contains a, and is perpendicular to b . the ellipse also has many rotational symmetries about b and $180°$ rotational symmetry about the many lines perpendicular b and containing the intersection of axes.