The problem:
An object is placed in front of a plane mirror and is displaced by 0.4 m along a straight line at an angle of 30 degrees to the mirror plane. We need to find the change in the distance between the object and its image.

Solution:

Understanding the scenario:
- A plane mirror forms virtual images.
- The image formed is the same distance behind the mirror as the object is in front of it.
- The image formed is laterally inverted.
- The angle of incidence is equal to the angle of reflection.

Initial scenario:
- Let's assume the object is placed at point O in front of the mirror.
- The distance between the object and the mirror is represented as d.
- The image of the object is formed at point I, which is the same distance behind the mirror as the object is in front of it.

Displacement of the object:
- The object is displaced by 0.4 m along a straight line at an angle of 30 degrees to the mirror plane.
- Let's assume the new position of the object is at point O'.
- The distance between the new position of the object and the mirror is represented as d'.

Calculating the change in distance:
- We can divide the displacement of the object into two components: one parallel to the mirror plane and one perpendicular to it.
- The component parallel to the mirror plane will be equal to d' - d.
- The component perpendicular to the mirror plane will be equal to d * sin(30°).
- The change in distance between the object and its image will be equal to the component perpendicular to the mirror plane, as the component parallel to the mirror plane does not affect the distance between the object and its image.
- Therefore, the change in distance is equal to d * sin(30°).

Calculating the change in distance:
- Given that the initial distance d between the object and the mirror is not mentioned, we cannot calculate the exact change in distance.
- However, we can calculate the change in distance relative to the initial distance.
- The change in distance as a percentage relative to the initial distance is equal to (d * sin(30°)) / d * 100%.
- Simplifying this expression, we get sin(30°) * 100% which is approximately 50%.

Conclusion:
- The change in the distance between the object and its image, relative to the initial distance, is approximately 50%.

Let the straight line be inclined at (pi/6) to normal from object to mirror. Change in distance between object and mirror=

0.4cos(pi/6), change in distance between image and mirror=-0.4cos(pi/6),so change in distance between object and image=

0.4cos(pi/6)-(-0.4cos(pi/6))=0.4*√3