A converging lens and a diverging mirror are placed at a separation of 15 cm. The focal length of the lens is 25 cm and that of the mirror is 40 cm. Where should a point source be placed between the lens and the mirror so that the light, after getting reflected by the mirror and then getting transmitted by the lens, comes out parallel to the principal axis ?

Explanation

To solve the problem step-by-step, we need to find the position of a point source placed between a converging lens and a diverging mirror such that the light after reflecting from the mirror and passing through the lens comes out parallel to the principal axis.

### Step 1: Understand the Setup
We have:
- A converging lens with a focal length \( f_L = +25 \) cm.
- A diverging mirror with a focal length \( f_M = -40 \) cm (negative because it is a diverging mirror).
- The distance between the lens and the mirror is \( d = 15 \) cm.

### Step 2: Identify the Condition for Parallel Rays
For the light rays to emerge parallel after passing through the lens, they must originate from the focus of the lens. Therefore, we need to find the position of the image formed by the mirror, which will act as the object for the lens.

### Step 3: Set Up the Coordinate System
Let:
- The position of the point source from the mirror be \( u \).
- The position of the image formed by the mirror be \( v \).

Since the mirror is a diverging mirror, the image formed will be virtual and located behind the mirror.

### Step 4: Use the Mirror Formula
The mirror formula is given by:
\[ \frac{1}{f_M} = \frac{1}{v} + \frac{1}{u} \]
Where:
- \( f_M = -40 \) cm (focal length of the mirror)
- \( v = +10 \) cm (the image is virtual and located 10 cm behind the mirror)

Rearranging the formula gives:
\[ \frac{1}{u} = \frac{1}{f_M} - \frac{1}{v} \]

### Step 5: Substitute the Values
Substituting the known values:
\[ \frac{1}{u} = \frac{1}{-40} - \frac{1}{10} \]
Calculating the right side:
\[ \frac{1}{u} = -\frac{1}{40} - \frac{4}{40} = -\frac{5}{40} = -\frac{1}{8} \]
Thus,
\[ u = -8 \text{ cm} \]

### Step 6: Find the Distance from the Lens
The distance of the point source from the mirror is \( |u| = 8 \) cm. Since the distance between the lens and the mirror is 15 cm, the distance of the point source from the lens \( d_L \) is:
\[ d_L = 15 - |u| = 15 - 8 = 7 \text{ cm} \]

### Conclusion
The point source should be placed 7 cm from the lens (or 8 cm from the mirror) for the light to come out parallel to the principal axis after reflecting from the mirror and passing through the lens.

### Summary of Results
- Distance of point source from the mirror: 8 cm
- Distance of point source from the lens: 7 cm