# Sliding Ladder

**Middle School Problem Exercises 2 and 3**2. A ladder leans against a vertical wall. A cat is seated on its central rung (i.e., at the midpoint of the ladder). The ladder, for some reason, starts to slip away from the wall. The cat stays on the central rung. What is the locus described by the cat? 3. The same question as Q2, but this time the cat is not seated at the midpoint of the ladder. What is the locus described by the cat in this situation?

## Questions that can guide you to visualize the locus:

i) If A(*a*,0) is a point on the *x*-axis and B (0,*b*) is a point on the *y-*axis and AB = 10, then what is the
relationship between *a* and *b*?

## Understanding the relationship between A and B

ii) When B is at the origin, what are the co-ordinates of A? And when A is at the origin, what are the co-ordinates of B?

## Visualising the movement of the ladder

iii) If A moves along the positive direction of the *x*-axis, what is the direction of motion
of B?

## And where is the cat on the ladder?

iv) What are the co-ordinates of the mid-point M of AB in both the instances in question (ii)?
What are the co-ordinates of the mid-point M of AB in terms of *a* and *b*?

v) What is the distance of M from the origin? Give your answer in terms of *a* and *b.
* What is the numerical value of this distance? Hint: Use the answer to question (i).

vi) What can you say about the distance of M from the origin as A (and therefore B) move?
Predict the locus of M as A moves along the *x*-axis.

vii) What changes if you consider the locus of any point on AB, instead of the mid-point? Right click on either of the other points shown on the ladder and choose Show Trace. Describe your observations.