Here is a question i am not sure how to tackle, I am not familiar with how to deal with family of curves and don't really have much time to look around for the definition as i am sitting the exam in two days.

The question is divided into three parts: Here is my attempt any help appreciated.

1) I am have no idea, i think it is a case of knowing the definition and i don't.

2) It is simply constraining the local parametrization to the given function so:

which is a constant.

3) is like phi therefore the tangent vectors to the family defined by the psi are of the multiples of

So for the families to be orthogonal their tangent must be orthogonal and so

Using the fundamental forms E=1=G and F=0 we get which after differentiating gives

And after that i am stuck ... any help would be appreciated.