Analytic Geometry: Orthogonal Trajectories

The problem is as follows:

a) Find the family of circles centered on the $y$-axis, that pass through the points $(\pm a,0)$, where [math]0<a\le r\in\mathbb{R}[/math].

b) Find the family of curves orthogonal to the family of circles found in part a).

Hint 1: Express the family of circles from part a) in the form:

[math]F(x,y)=C[/math]

Hint 2: It can be shown that the orthogonal trajectories must satisfy:

[math]\frac{\partial F}{\partial y}dx-\frac{\partial F}{\partial x}dy=0[/math]

Minimize The Crease

Consider a rectangular piece of paper of width $W$ laid on a flat surface. The lower left corner of the paper is bought over to the right edge of the paper, and the paper is smoothed flat creating a crease of length $L$, as in the diagram:

What is the minimal value of $L$ in terms of $W$?