$(a)$ Define power of a lens and write its $SI$ unit.
$(b)$ $A$ convex lens forms a real and inverted image of a needle at a distance of $50\, cm$ from it. Where is the needle placed in front of the lens, if the image size is equal to the object size? Also, find the power of the lens.

$(D)$ The power of a lens is defined as the reciprocal of its focal length in meters. It represents the ability of a lens to converge or diverge rays of light. Its $SI$ unit is dioptre $(D)$.
$(b)$ Given that the image is real, inverted, and of the same size as the object, the object must be placed at $2f$ (twice the focal length) from the lens, and the image is also formed at $2f$ on the other side.
Given $v = 50\, cm$, therefore $2f = 50\, cm$, which implies $f = 25\, cm = 0.25\, m$.
The needle is placed at $50\, cm$ in front of the lens.
The power of the lens $P = \frac{1}{f(\text{in meters})} = \frac{1}{0.25} = +4\, D$.