Find Focal Length from Magnification and Distance

 

❓ Question

A mirror produces an image with a magnification of 1/4.
If the distance between the object and its image is 40 cm, find the focal length of the mirror.

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🖼️ Question Image

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✍️ Short Explanation

This problem is based on:

👉 Mirror formula
👉 Magnification
👉 Image formation by mirrors.

Main idea:

For mirrors:

$$\boxed{ m=-\frac{v}{u} }$$

and:

$$\boxed{ \frac1f=\frac1v+\frac1u }$$

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🔷 Step 1 — Use Magnification Formula 💯

Given:

$$m=\frac14$$

For mirrors:

$$m=-\frac vu$$

Thus:

$$-\frac vu=\frac14$$
$$v=-\frac u4$$

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🔷 Step 2 — Use Object-Image Distance

Distance between object and image:

$$40\text{ cm}$$

Since image is real and diminished:

✔ Object and image lie on same side of mirror.

Thus separation:

$$|u|-|v|=40$$

Using:

$$v=\frac u4$$
$$u-\frac u4=40$$
$$\frac{3u}{4}=40$$
$$u=\frac{160}{3}\text{ cm}$$

Then:

$$v=\frac{40}{3}\text{ cm}$$

Using sign convention:

$$u=-\frac{160}{3},\quad v=-\frac{40}{3}$$

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🔷 Step 3 — Apply Mirror Formula

$$\frac1f=\frac1v+\frac1u$$

Substitute:

$$\frac1f = -\frac3{40} -\frac3{160}$$

Take LCM:

$$= -\frac{12+3}{160}$$
$$= -\frac{15}{160}$$
$$= -\frac3{32}$$

Thus:

$$f=-\frac{32}{3}$$

Magnitude:

$$|f|=10.67\text{ cm}$$

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🔷 Step 4 — JEE Trap Alert 🚨

❌ Magnification sign ignore kar dena

❌ Object-image distance directly add kar dena

❌ Mirror sign convention bhool jaana

Remember:

For diminished real image by concave mirror:

$$\boxed{ m<1 }$$

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✅ Final Answer

$$\boxed{ 10.7\text{ cm} }$$

(Option 4)

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