Chapter 9.2, Problem 69E

(a)

To determine

To find: The length of the string in terms of $a$ .

Answer to Problem 69E

The length of the string in terms of $a$ is $2a$ .

Explanation of Solution

Given information:

An ellipse can be drawn using two thumbtacks, a string of fixed length and a pencil. When the ends of the string are fastened, all the tacks and the string is drawn taut with a pencil. The path traced by the pencil is ellipse.

Calculation:

Assume that two thumbtacks are the foci of any ellipse and represented by ${\text{F}}_{\text{1}}{\text{ and F}}_{\text{2}}$ .

  $\text{P}$ is any point on the path traced by the pencil. Consider $\text{P}$ is at a particular position on the path as shown in the figure.

  

From figure, center of foci is $\text{O}$ . Let us suppose that:

  $\begin{array}{l}a=\text{Distance between O and P}\\ c=\text{Distance of O to the foci}\text{.}\end{array}$

So,

  $\begin{array}{l}\text{OP}=a\\ {\text{OF}}_1=c\\ {\text{OF}}_{\text{2}}=c\\ {\text{PF}}_{\text{2}}=\text{OP}-{\text{F}}_{\text{2}}\text{P}\\ =a-c\end{array}$

Calculate the length of the string $l$ .

  $\begin{array}{l}l={\text{OF}}_1+\text{OP}+{\text{PF}}_{\text{2}}\\ =\cancel{c}+a+a-\cancel{c}\\ =2a\end{array}$

Therefore, the length of the string in terms of $a$ is $2a$ .

(b)

To determine

To explain: The reason for path to be an elliptical curve.

Answer to Problem 69E

The path traced by pencil is elliptical.

Explanation of Solution

Given information:

An ellipse can be drawn using two thumbtacks, a string of fixed length and a pencil. When the ends of the string are fastened all the tacks and the string is drawn taut with a pencil, the path traced by the pencil is ellipse.

The given activity satisfies the definition of an ellipse that is an ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant.

Therefore, the path traced by pencil is elliptical.