Question

An object placed at a distance of 9cm from the first principal focus of a convex lens, produces a real image at a distance of $25cm$ from its second principal focus. Then focal length of the lens is:
A) $9cm$
B) $25cm$
C) $15cm$
D) $17cm$

Answer 1

Using the mirror formula proceed with the given data. All the given data are in terms of focus, thus the object and image distance must be calculated in terms of focal length.

Complete step by step answer:
Let f be the focal length of the given convex lens.
We know, according to the new sign conventions, the following signs can be considered:
Since, we have a convex lens, the focal length lies on the left side of the lens and hence it is taken to be positive.
Object is always kept in the left hand side of the lens, thus object distance is also taken in negative.
Image is formed on the right hand side, thus has a positive sign.
As given in the question,
Let us consider:
$u =$Object Distance
$v =$Image Distance
As given in the question:
$u = - (9 + f)$
$v = + (25 + f)$
Now, applying the Lens formula:
$\dfrac{1}{f} = \dfrac{1}{v} - \dfrac{1}{u}$
Now, putting the values are mentioned above:
$\dfrac{1}{f} = \dfrac{1}{{ + (25 + f)}} - \dfrac{1}{{ - (9 + f)}}$
On solving the above equation, we obtain:
${f^2} + 25f + 9f + 225 = 2{f^2} + 34f$
From, here we obtain the unknown value:
${f^2} = 225cm$
$\Rightarrow f = 15cm$
Since, the focal length of a convex lens is positive. Thus:
$f = 15cm$
This is our required answer.

Therefore, option (C) is correct.

Note: The signs of image distance, object distance and focal length must be considered according to the New Sign Convention. Considering the lens formula, if any two quantities from image distance, object distance and focal length if known, the other can be found.