A point object is placed on the principal axis of a concave mirror quite far away from the pole and moved at a constant speed `0.5 cm//sec` towards the pole. Its image also moves. It is found that the object and the image cross each other at a point which is at a distance 50 cm from the pole. Focal length of the mirror is -

Options

A. 50 cm
B. 35 cm
C. 25 cm
D. 15 cm

Answer

To find the focal length of the concave mirror given the conditions of the problem, we can follow these steps:

### Step 1: Understand the setup
We have a point object placed far away from the concave mirror and moving towards it at a constant speed of 0.5 cm/sec. The image of the object also moves, and they cross each other at a point 50 cm from the pole of the mirror.

### Step 2: Use the mirror formula
The mirror formula relates the object distance (u), image distance (v), and focal length (f) of a mirror:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]

### Step 3: Identify distances
When the object and image cross at a distance of 50 cm from the pole, we can denote this distance as:
- \( v = -50 \) cm (the image distance is negative for concave mirrors)
- The object distance \( u \) will be equal to \( -2f \) when they meet at this point.

### Step 4: Set up the equation
Since the object and image cross each other at 50 cm from the pole, we can set:
\[ u = -50 \text{ cm (when they cross)} \]
Using the mirror formula:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]
Substituting \( u = -50 \) cm and \( v = -50 \) cm:
\[ \frac{1}{f} = \frac{1}{-50} + \frac{1}{-50} \]
\[ \frac{1}{f} = -\frac{1}{50} - \frac{1}{50} = -\frac{2}{50} = -\frac{1}{25} \]

### Step 5: Solve for focal length
Taking the reciprocal gives us:
\[ f = -25 \text{ cm} \]

### Conclusion
The focal length of the concave mirror is \( -25 \) cm.

---