Introduction:
Focal length is an important property of lenses and mirrors which determines how they refract or reflect light. However, the sign convention for focal length differs for concave lenses and convex mirrors. This is because of their respective geometries and the way they form images.

Focal length of concave lens:
A concave lens is thinner at the center than at the edges, and it causes light rays to diverge. When parallel rays of light pass through a concave lens, they diverge as if coming from a point behind the lens. This point is called the virtual focus or the focal point. Since this point is on the opposite side of the lens from where the light is coming, the focal length of a concave lens is negative.

Focal length of convex mirror:
A convex mirror is curved outward like the back of a spoon, and it causes light rays to reflect outwards. When parallel rays of light reflect off a convex mirror, they appear to come from a point behind the mirror. This point is called the virtual focus or the focal point. Since this point is on the same side of the mirror as the incoming light, the focal length of a convex mirror is positive.

Sign convention for focal length:
The sign convention for focal length is based on the direction in which light is coming and going. When light is coming from the left and going to the right, the focal length is positive. When light is coming from the right and going to the left, the focal length is negative.

Conclusion:
In summary, the sign convention for focal length differs for concave lenses and convex mirrors because of the way they refract or reflect light. A concave lens causes light to diverge and its focal length is negative, while a convex mirror causes light to reflect outward and its focal length is positive.

The focal length, is positive for a concave mirror, and negative for a convex mirror. When the image distance is positive, the image is on the same side of the mirror as the object, and it is real and inverted. When the image distance is negative, the image is behind the mirror, so the image is virtual and upright.