Question

how to derive the magnification formula
M= hi/ho=di/do

Answer 1

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Answer 2

In general we have the relation
1/f = 1/u + 1/v where f is the focal length, u is the distance of the object from the optic center, and v is the distance of the image from the optic center.

If ho is the height of the object from the principle axis, and hi is the height of the image, then magnification m = hi/ho. From similar triangles, hi/ho = v/u.
Using this in the general formula 1/u+1/v = 1/f,
u/v = (u/f )-1 = [u -f] /f or
m = f/[u-f]
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Now we apply the appropriate sign conventions for concave-convex - mirrors and lenses in the above equation.