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The image formed by a convex mirror of focal length $ 30 \mathrm{cm} $ is the quarter of size of the object. Then the distance of the object from the mirror is,
(A) $ 30 \mathrm{cm} $
(B) $ 90 \mathrm{cm} $
(c) $ 120 \mathrm{cm} $
(D) $ 60 \mathrm{cm} $

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Last updated date: 17th Jun 2024
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Answer
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Hint
We know that a convex mirror is a spherical reflecting surface or any reflecting surface fashioned into a portion of a sphere in which its bulging side faces the source of light. Automobile enthusiasts often call it a fish eye mirror while other physics texts refer to it as a diverging mirror. Convex mirrors always form images that are upright, virtual, and smaller than the actual object. They are commonly used as rear and side view mirrors in cars and as security mirrors in public buildings because they allow us to see a wider view than flat or concave mirrors. Convex mirrors are used inside the buildings, they are also used in making lenses of sunglasses, they are used in the magnifying glass, they are used in securities and they are used in telescopes.

Complete step by step answer
We know that, The focal length of the convex mirror is $ \mathrm{f}=30 \mathrm{cm} $ and the magnification of the image is $ \mathrm{m}=1 / 4 $
We have to calculate the object distance u. According to sign convention, the measurements along the direction of light are taken as positive and opposite to the light taken as negative. The transverse measurement above the principal axis is taken as positive and below the principal axis is taken as negative. Thus, $ f=+30 \mathrm{cm} $ and $ \mathrm{m}=+1 / 4=+0.25 $
Using formula, $ \mathrm{m}=\dfrac{\mathrm{f}}{\mathrm{f}-\mathrm{u}} $
or $ \mathrm{mf}-\mathrm{mu}=\mathrm{f} $
or $\mathrm{u}=\dfrac{\mathrm{mf}-\mathrm{f}}{\mathrm{m}}=\dfrac{\mathrm{m}-1}{\mathrm{m}} \mathrm{f}=\dfrac{0.25-1}{0.25}(30)=-90 \mathrm{cm} $
The minus sign indicates that we had assumed a wrong direction.

Therefore, the correct answer is Option (B).

Note
It should be known that in a thin lens in air, the focal length is the distance from the center of the lens to the principal foci (or focal points) of the lens. For a converging lens (for example a convex lens), the focal length is positive, and is the distance at which a beam of collimated light will be focused to a single spot. The focal length of the lens is the distance between the lens and the image sensor when the subject is in focus, usually stated in millimetres (e.g., 28 mm, 50 mm, or 100 mm). In the case of zoom lenses, both the minimum and maximum focal lengths are stated, for example 18–55 mm. Focal length can also change the perspective and scale of your images. A lens with a shorter focal length “expands” perspective, giving the appearance of more space between the elements in your photo. Meanwhile, telephoto lenses tend to stack elements in the frame together to “compress” perspective.