Hi so I've been given a question for a Maths assignment in relation to conics and its applications. The question is:
A $6m$ ladder lies against a wall. Its bottom is pulled along the floor away from the wall. Taking the coordinate axis along the floor and the wall, it can be shown that the locus of the ladder $2m$ from the top of the ladder forms an ellipse.
(People are probably going to get annoyed at me for this but there is 2 parts to the question)
a) find the locus of the point described above.
b) Refine this model to find the general locus if the point is at a point $tm$ from the top of a ladder $p$ meters long and then use it to find the locus for a ladder of length of $7m$ and a point $3.2m$ from the top.
If anyone could explain the locus and the general equation that would be all i need and i would be so greatful, Thankyou! :)
Let the ladder have length $p$ and the point which is $t$ units from the top of the ladder have coordinates $(x,y)$.
Now let $\theta$ be the angle between the ladder and the horizontal.
A quick diagram will comfirm that $$x=t\cos\theta$$ and $$y=(p-t)\sin\theta$$
Eliminating $\theta$ gives the equation of the locus of $(x,y)$ which is the ellipse $$\frac {x^2}{t^2}+\frac {y^2}{(p-t)^2}=1$$