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.
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.
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