Question

why the focal length of convex lens is always positive?

Answer 1

I have a great deal of sympathy with your position. As taught in schools the sign convention is somewhat vague and that gets confusing with complicated setups.

If you're dealing with anything complicated I recommend keeping strictly to the Cartesian sign convention. I've linked an article that seems to be a good summary of this, but Googling will find you lots more articles to look at. Basically:

the light travels from left to right

to the left of the lens is negative

to the right of the lens is positive

converging lenses have positive ff, diverging lenses have negative ff

This convention means uu is normally negative because the object is to the left of the lens, while vv is normally positive because a (real) image is to the right of the lens. The lens equation becomes:

1/v−1/u=1/f

I HOPE THIS HELPS U ✅
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Answer 2

Convex lens converges incident rays. Hence the rays to converge at a point in front of the lens. So image distance is positive here. We can easily clarify that the focal length is positive direction for a Convex lens while negative with concave lens.