Magnification M is defined as :
M = - q / p
(*where q=image distance, p=object distance)
we can further solve this equation for q, (which is needed later on).
q = - M*p
Based on the properties on lens, we can conclude that for M<1 it is a diverging lens, and the image is virtual.
Here are some basic sign conventions for reference:
the distance between the object and its image is given by d.
For M < 1, we will have
d = p + q
substituting in q we get:
d = p + -M*p
solving for p we get:
p = d / (1-M)
We can now plug in p and q into the thing lens equation to solve for f:
1/p + 1/q = 1/f
(1-M)/ d + 1/-M*p = 1/f
solving for f we get:
The steps for solving M>1 is pretty much the same.
only this time you have a converging lens, and the image is still virtual. So we get that
d = -q - p
All the other steps are exactly the same as earlier. the final solution will be: