Finding the index of refraction with respect to the vertical axis

[greedo]

Homework Statement

Studying the mirage, we can assume that the index of refraction depends only on the vertical coordinate z. A ray of light starts from {0,0} with angle $$\beta\leq30$$. It's trajectory fits on an ellipse with major axis a and minor axis b.

a) Give $$n(z)=?$$
b) What is the trajectory of a ray of light that starts with angle $$2\beta$$ ?

Homework Equations

From functional analysis we derived that in this situation:
$$\frac{n[z]}{\sqrt{1+z'[x]^2}}=\text{constant}$$

The Attempt at a Solution

What I have problems with is expressing u and v in terms of beta,a and b. If I could do that then I could give the equation for the ellipse, order it for z[x] and insert it into the equation in 2. I was supposed to do this in an exam in about half an hour, so it has to have a simple solution that I cannot see even after looking at it for hours.

I have made a drawing

 

[Redbelly98]

Welcome to Physics Forums.

Are u and v the (x,z) coordinates of the center of the ellipse? What about starting with the general equation for an ellipse, and setting z'=tan(β) at the origin?

 

[greedo]

Yes they are. So I've ordered the general equation for z, derived it, equated with tan(beta):

http://img268.imageshack.us/img268/6720/32720014.th.png

and solved for u and v:

http://img138.imageshack.us/img138/4112/60858362.th.png

and inserting back, I've got:

http://img138.imageshack.us/img138/2965/12957190.th.png

Do you think that's correct? Thank you for the suggestion.