Optical centre and converging lens numerical on meter scale

Optical centre is the central point of a lens through which a ray of light passes without suffering any deviation.

• Object position = $8\text{ cm}$, Lens position = $50\text{ cm} \implies u = 8 - 50 = -42\text{ cm}$.
• Screen (Image) position = $92\text{ cm}$, Lens position = $50\text{ cm} \implies v = 92 - 50 = +42\text{ cm}$. Using lens formula: $\frac{1}{f} = \frac{1}{v} - \frac{1}{u} = \frac{1}{42} - \left(-\frac{1}{42}\right) = \frac{2}{42} = \frac{1}{21}$ $f = +21\text{ cm}$

• New object position = $29\text{ cm} \implies u' = 29 - 50 = -21\text{ cm}$. Since $u' = -f = -21\text{ cm}$, the object is at the principal focus $F_1$. Using lens formula: $\frac{1}{v'} = \frac{1}{f} + \frac{1}{u'} = \frac{1}{21} - \frac{1}{21} = 0 \implies v' = \infty$ The image is formed at infinity.

If the object is shifted closer than $29\text{ cm}$ mark (distance from lens $< 21\text{ cm}$), the object distance becomes less than the focal length ($|u| < f$). The image formed will be virtual, erect, and magnified.