Panoramic Mirror with Infinity Trick

Not to sound lazy or anything but the numeric solution for the panoramic mirror in the last post was just too much work!  That is why we tried to solve it analytically, this time.
So, we still have the same panoramic mirror as before, where the object is positioned on a surrounding wall with the coordinates (k, ax+b). The image plane is below the x-axis and is indicated by y=-k.

But, here is the time to use a trick.  We know that "a" is an arbitrary scaling factor and the bigger the "a", the bigger the field of view.  Therefore, it shouldn't matter whether it is ax+b or kax+b or gax+b.  "a", "ka" and/ or "ga" are all scaling factors.  So, for the sake of our trick, let's say it is kax+b. 

Now, once again, we draw the tangent line to the mirror at point (x, F(x)) and we have:

(1)

 (2)

Using the tangent double-angle formula we have:  

 (3)

Here is where the trick comes into the picture.  Suppose k is going to infinity and find the limit of tangent of 2theta:

 (4)

It looks so cool, doesn't it?

By equating the (2) and the (4) equations, we get:

YAY! We got our differential equation.  Now, by solving this differential equation in maple and finding the surface of revolution in Pov-ray we obtain the same results as we did previously :)