From a convex mirror of focal length 3F0 an object is placed at a distance of 2F0 as shown in the figure. Where should we place a plane mirror in between object and convex mirror so that image formed by both the mirrors is formed at the same location. Find the distance of plane mirror from object.

To find the distance of the plane mirror from the object, we need to consider the properties of convex mirrors and plane mirrors.

1. Given that the object is placed at a distance of 2F0 from the convex mirror, we can determine the focal length of the convex mirror as F0/3.

2. The image formed by the convex mirror will be virtual, erect, and diminished in size.

3. To have the image formed by both the convex mirror and the plane mirror at the same location, we need to ensure that the virtual image formed by the convex mirror is reflected by the plane mirror and appears to be coming from the same point.

4. Since the image formed by the convex mirror is virtual, the reflected image from the plane mirror will also be virtual.

5. For the virtual image to appear to be coming from the same point, the plane mirror should be placed at a distance equal to the distance between the convex mirror and the virtual image formed by it.

6. The distance between the convex mirror and the virtual image can be calculated using the mirror formula:

   1/f = 1/v - 1/u

   where f is the focal length of the convex mirror, v is the image distance, and u is the object distance.

7. Substituting the given values, we have:

   1/(F0/3) = 1/v - 1/(2F0)

   Simplifying, we get:

   3/F0 = 1/v - 1/(2F0)

8. Rearranging the equation, we have:

   1/v = 3/F0 + 1/(2F0)

   1/v = (6 + 1)/(2F0)

   1/v = 7/(2F0)

   v = 2F0/7

9. Therefore, the distance between the convex mirror and the virtual image is 2F0/7.

10. To have the image formed by both mirrors at the same location, the plane mirror should be placed at the same distance, which is 2F0/7, from the object.

Thus, the distance of the plane mirror from the object is 2F0/7.