Chapter 14.8, Problem 1C

To determine

To describe the locus of the cat as the mouse moves from A to J.

Answer to Problem 1C

The locus of the cat is only at point C .

Explanation of Solution

Given information:

A mouse moves along $\overline{AJ}$ . For any position M of the mouse, X and Y are such that $\overline{AX}\perp \overline{AJ}$ with $AX=AM$ , and $\overline{JY}\perp \overline{AJ}$ with $JY=JM$ . The cat is at C , the midpoint of $\overline{XY}$ .

The locus of the cat is only at point C . since $\overline{AX}\perp \overline{AJ}$ and $\overline{JY}\perp \overline{AJ}$ it means that AX is parallel to JY . From the given condition it is clear that at the start when mouse is at point A,AX is equals to zero and JY is equal to AJ . As the mouse moves forward the length of AX increases and the length of JY decreases and at the end point J,AX is equal to AJ and JY becomes zero. Hence, the line XY rotates about its midpointC . therefore the point C never changes its coordinates.

Hence, the locus of the cat is only at point C .