Habluka's Weekly Challenge #2

[Habluka]

Since there's another thread with relatively simple problems that people are posting in general, I've decided to make the ones in my threads slightly more interesting now, which means it's possible that more people will not be able to solve them. Also, solutions from writing programs will be harder to get now.

Anyway, I would like to congratulate the following people for getting correct answers for one or more of last week's problems: Aekorn, Sarath, Sephr, Lantry (even though I don't like his answer), and also Roflcopter even though he didn't actually give an answer (his post was relevant enough to tell he knew the answer)

Now for this week's problems:

1. • For a given parametric equation (x(t), y(t)), there is a parametric equation (x1(t), y1(t)) where for each value of t, (x1(t), y1(t)) is the end point of a perpendicular line segment to the first equation with a starting point (x(t), y(t)) and a length d. Assuming d(t) is a constant and both equations can be written as functions of y in terms of x, what is the area of the shape you get if you trace the movement of the line segment between (x(t), y(t)) and (x1(t), y1(t))?
     
   • In the case where x(t) = t and y(t) = sin(t), what is the maximum value of d for which an area can be found using this method?
     Here's an image that should help visualize what sort of thing this problem is asking
     Spoiler

     
   
   
2. Let f(x) be a linear function and let Vector V have a magnitude m and a direction theta. Find the orthogonal projection of V onto f(x).