From the thin lens formula:
$$\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$$
Multiplying through by $v$:
$$1 - \frac{v}{u} = \frac{v}{f}$$
Since magnification $m = \frac{v}{u}$, this gives:
$$1 - m = \frac{v}{f} \implies m = 1 - \frac{v}{f}$$
Taking the magnitude/slope of the linear plot of $m$ versus $v$:
$$\left|\frac{dm}{dv}\right| = \frac{1}{f}$$
From the given graph:
- Vertical change (change in magnification, $\Delta m$) = $c$
- Horizontal change (change in image distance, $\Delta v$) = $b$
Therefore, the slope of the line is:
$$\text{Slope} = \frac{\Delta m}{\Delta v} = \frac{c}{b}$$
Equating slope to $\frac{1}{f}$:
$$\frac{1}{f} = \frac{c}{b} \implies f = \frac{b}{c}$$
Hence, option 4 is the correct answer.
पतले लेंस के सूत्र से:
$$\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$$
दोनों पक्षों में $v$ से गुणा करने पर:
$$1 - \frac{v}{u} = \frac{v}{f}$$
चूंकि आवर्धन $m = \frac{v}{u}$ होता है:
$$1 - m = \frac{v}{f} \implies m = 1 - \frac{v}{f}$$
अतः $m$ और $v$ के मध्य खींची गई सरल रेखा की प्रवणता (ढाल):
$$\text{Slope} = \frac{\Delta m}{\Delta v} = \frac{1}{f}$$
दिए गए ग्राफ़ से:
- ऊर्ध्वाधर परिवर्तन ($\Delta m$) = $c$
- क्षैतिज परिवर्तन ($\Delta v$) = $b$
अतः प्रवणता = $\frac{c}{b}$:
$$\frac{1}{f} = \frac{c}{b} \implies f = \frac{b}{c}$$
अतः विकल्प 4 सही उत्तर है।
