OPTICS: URGENT QUESTION-- DETAILS PLEASE
What's the paraxial approximation, in spherical mirrors? start from the equation \left(x-R\right)^2+y^2=R^2(x−R) 2 +y 2 =R 2 , which represents a circle of radius RR with the center at x=R, and solve for x. Now use Taylor expansion to show that the circle can be approximated as parabola, for small y's. Find the relationship between the radius and focal length.
leaving aside the formatting garbage, you have
(x-r)^2 + y^2 = r^2
(x-r)^2 = r^2 - y^2
x = r + √(r^2-y^2)
Now just expand that to a series and ignore everything past y^2.