Chapter A5.1, Problem 46E

To determine

To show: why the pencil traces a parabola.

Answer to Problem 46E

The pencil traces a parabola because the length of the string is unchanged.

Explanation of Solution

Given:

  

Calculation:

This happens because the length of the string is unchanged. This is so because the string is inextensible.

Here,

  $\begin{array}{l}AB=FP+PA\\ =AP+PB\end{array}$

Assume at the beginning, $AF=AB$

Now, as the T square comes closer to the focus point $(F),$ the length AP increases. Assume an incremental increase in the length (of x) happens in APx $)happensin$ A P$.

Now the distance between the point $B$ and the pencil becomes $AB-x$ .

And correspondingly, the string length would reduce to $AF-x$ .

According to our assumption, $AB-x=AF-x$

This is nothing but the locus of a parabola (locus of a point that moves such that the distance of the point from a fixed point is equal to its distance from a fixed line).

Conclusion:

Therefore, the pencil traces a parabola because the length of the string is unchanged.