Chapter 8, Problem 77RE

Solution

a)

To prove: The ball hit a blocked spot on Elliptical Billiard Table.

Given information:

Major axis is 6 ft and minor axis is 4 ft.

Calculation:

Ellipse with Major axis is 6 and minor axis is 4.

  

Figure (1)

The shot passing through one focus in the ellipse is reflected through other focus.If first shot misses the focus its never go through the focus consequently.

So, the shot is targeted through one of the foci to reflected through other foci.

b)

To find:the point(s) for which the ball is aimed to shot.

The ball is aimed to shot through the focal points $\left(-2.82,0\right),\left(2.82,0\right)$ .

Given information:Major axis is 6 ft and minor axis is 4 ft.

Formula used:

If the semi-major axis length is $a$ and semi-minor axis length is $b$ then the eccentricity is $e=\sqrt{1-\frac{b^2}{a^2}}$ and Foci are $\left(\pm ae,0\right)$ .

Calculation:

Eccentricity:

Substitute the value 3 for $a$ and 2 for $b$ in $e=\sqrt{1-\frac{b^2}{a^2}}$ .

  $\begin{array}{c}e=\sqrt{1-\frac{b^2}{a^2}}\\ =\sqrt{1-\frac{4}{36}}\\ =\sqrt{\frac{32}{36}}\\ =0.94\end{array}$

Thus, the eccentricity of the given ellipse is 0.94.

Foci:

Substitute the value 3 for $a$ and $0.94$ for $e$ in $\left(\pm ae,0\right)$ .

The foci are

  $\begin{array}{c}\left(\pm ae,0\right)=\left(\pm 3\left(0.94\right),0\right)\\ =\left(\pm 2.82,0\right)\end{array}$

Therefore, the foci are $\left(-2.82,0\right),\left(2.82,0\right)$ .

So, the ball is aimed to shot through the focal points $\left(-2.82,0\right),\left(2.82,0\right)$ .