Question

The same size images are formed by a convex lens when the object is placed at 20 cm or at 10 cm from the lens. The focal length of convex lens is _________ cm.

Hint:

If an object at $u_1$ and $u_2$ produces images of the same magnification, the focal length is the arithmetic mean of the two distances: $f = \frac{u_1+u_2}{2}$.

Answer 1

Correct Answer: 15

Step 1: For a convex lens, "same size image" at two different positions means one image is real ($m = -k$) and the other is virtual ($m = +k$).
Step 2: Magnification formula: $m = \frac{f}{f+u}$.
Step 3: For $u_1 = -20$: $m_1 = \frac{f}{f-20}$.
Step 4: For $u_2 = -10$: $m_2 = \frac{f}{f-10}$.
Step 5: Since $|m_1| = |m_2|$ and one must be negative: $\frac{f}{f-20} = -\frac{f}{f-10}$.
Step 6: $f-10 = -(f-20) \Rightarrow f-10 = -f+20 \Rightarrow 2f = 30 \Rightarrow f = 15$ cm.