Question

An object is placed at a distance of 75 cm from a screen. A convex lens of focal length 12 cm is placed in between. The position of the lens with respect to object such that a real image will form on the screen is
A. 15 cm
B. 60 cm
C. 5 cm
D. Both (1) and (2)

Answer 1

Hint: The convex lens has two values of the object for which the real image is produced on the screen. Use lens formula to determine the object distance.

We can use the Formula:
\[\dfrac{1}{f} = \dfrac{1}{v} - \dfrac{1}{u}\]
Here, $f$ is the focal length of the lens, $v$ is the object distance and $u$ is the image distance.


Step by step answer: Let the distance between the lens and object be\[x\]. Therefore, the distance between the lens and screen is \[\left( {75 - x} \right)\,cm\].

Use the lens formula to determine the image distance as follows,
\[\dfrac{1}{f} = \dfrac{1}{v} - \dfrac{1}{u}\]
Here, $f$ is the focal length of the lens, $v$ is the object distance and $u$ is the image distance.

The focal length of the convex lens is positive. Also, the object distance is negative and image distance is positive.

Substitute \[75 - x\] for v, \[ - x\] for u, and 12 cm for f in the above equation.
\[\dfrac{1}{{12\,cm}} = \left( {\dfrac{1}{{75 - x}}} \right) - \left( {\dfrac{1}{{ - x}}} \right)\]
\[ \Rightarrow \dfrac{1}{{12\,cm}} = \dfrac{{x + 75 - x}}{{\left( {75 - x} \right)x}}\]
\[ \Rightarrow \dfrac{1}{{900}} = \dfrac{1}{{75x - {x^2}}}\]
\[ \Rightarrow {x^2} - 75x + 900 = 0\]

Solve the above equation to get the values of $x$.
\[\left( {x - 15} \right)\left( {x - 60} \right) = 0\]
\[ \Rightarrow x = 15\,cm\,\,{\text{or}}\,\,60\,cm\]

So, the correct answer is option (D).

Note: The convex lens gives real image at two positions of the object with respect to the lens. Therefore, you should have gotten this that the answer will have two values. Remember, the distance towards the left of the lens is negative and the distance towards the right of the lens is positive.