Question

The diagram shows a ladder of length 2λ leaning against a wall so that the foot of the ladder is distant 2α from the wall. HELP!?

(a) Find the coordinates of B.

(b) Show that the midpoint P of the ladder lies on a circle with centre at the origin. What is the radius of this circle?

Answer 1

Let `O` be the origin `(0,0)`
So `OA=2alpha`

By Pythagoras theorem

`OB=sqrt(AB^2-OA^2)`

`=>OB=sqrt(4lambda^2-4alpha^2)`

`=>OB=2sqrt(lambda^2-alpha^2)`

So coordinates of `B` is `(0,2sqrt(lambda^2-alpha^2))`

And coordinates of `A` is `(2alpha,0)`

If the coordinates of mid point (`P`) of the ladder `AB` be `(h,k)` then

`h=(2alpha+0)/2=alpha`

`=>h=alpha.....(1)`

Again

`k=(2sqrt(lambda^2-alpha^2)+0)/2=sqrt(lambda^2-alpha^2)`

`=>k=sqrt(lambda^2-alpha^2).....(2)`

Combining (1) and (2) we get

`k=sqrt(lambda^2-h^2)`

`=>h^2+k^2=lambda^2`

Substituting `h=x and k=y` we get equation of locus of the mid point `(P)` of the ladder as

`x^2+y^2=lambda^2` which is obviously the equation of a circle having center `(0,0`) and radius `=lambda`