A real point object is kept on the pricipal axis of a concave mirror (radius of curvature `=20cm`) at a distance 60 cm from the pole. Take pole as origin and direction of incident light as positive direction of X axis. Y axis is perpendicular to the principal axis. Select correct statement(s).

Options

A. Option1 If velocity of object `t=1hati` and velocity of mirror `=0` then velocity of image will be `-(1)/(25)hati`
B. Option2 If velocity of object `=1hatj` velocity of mirror `=0` then velocity of image will be `-(1)/(5)hatj`
C. Option3 if velocity of object `=0` and velocity of mirror `=1hati` then velocity of image will be `(1)/(25)hati`
D. Option4 if velocity of object `=0` ad velocity of mirror `=1hatj` then velocity of image will be `(1)/(5)hatj`

Explanation

To solve the problem, we need to analyze the situation involving a concave mirror and the object placed in front of it. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the focal length of the concave mirror
The radius of curvature (R) of the concave mirror is given as 20 cm. The focal length (F) of a concave mirror is given by the formula:
\[ F = -\frac{R}{2} \]
Substituting the value of R:
\[ F = -\frac{20 \, \text{cm}}{2} = -10 \, \text{cm} \]

### Step 2: Determine the object distance (U)
The object is placed at a distance of 60 cm from the pole of the mirror. Since we take the pole as the origin and the direction of incident light as positive, the object distance (U) will be:
\[ U = -60 \, \text{cm} \quad (\text{real object is negative}) \]

### Step 3: Calculate the magnification (M)
The magnification (M) of a mirror is given by the formula:
\[ M = \frac{F}{F - U} \]
Substituting the values of F and U:
\[ M = \frac{-10}{-10 - (-60)} = \frac{-10}{-10 + 60} = \frac{-10}{50} = -\frac{1}{5} \]

### Step 4: Analyze the velocity of the object and image
1. **Case 1: Object moving in the x-direction**
- Let the velocity of the object with respect to the mirror be \( v_o = 1 \, \hat{i} \) (in the positive x-direction).
- The velocity of the image with respect to the mirror is given by:
\[ v_i = -M^2 \cdot v_o \]
Substituting the values:
\[ v_i = -\left(-\frac{1}{5}\right)^2 \cdot (1 \, \hat{i}) = -\frac{1}{25} \, \hat{i} \]

2. **Case 2: Object moving in the y-direction**
- Let the velocity of the object with respect to the mirror be \( v_o = 1 \, \hat{j} \) (in the positive y-direction).
- The velocity of the image with respect to the mirror is given by:
\[ v_i = -M \cdot v_o \]
Substituting the values:
\[ v_i = -\left(-\frac{1}{5}\right) \cdot (1 \, \hat{j}) = \frac{1}{5} \, \hat{j} \]

### Step 5: Evaluate the options
- **Option 1**: Velocity of object \( = 1 \hat{i} \), velocity of mirror \( = 0 \) → Velocity of image \( = -\frac{1}{25} \hat{i} \) (Correct)
- **Option 2**: Velocity of object \( = 1 \hat{j} \), velocity of mirror \( = 0 \) → Velocity of image \( = \frac{1}{5} \hat{j} \) (Correct)
- **Option 3**: Velocity of object \( = 0 \), velocity of mirror \( = -\hat{i} \) → Velocity of image calculation shows it is not correct.
- **Option 4**: Similar analysis shows it is not correct.

### Conclusion
The correct statements are:
- Option 1 and Option 2 are correct.