Show mathematically that to produce an image that is the same size as the object in a concave mirror, the object needs to be at twice the focal length from the mirror.

1/o + 1/o=1/f

1/2f + 1/i=1/f
1/i=1/2f
i=2f

M=di/do= 2f/2f= 1

check my thinking.

I'm sorry, but I'm not really understanding this; could you please xplain?

To mathematically derive the relationship between the object's position and the resulting image's size in a concave mirror, we can use the mirror formula and the magnification formula.

The mirror formula describes the relationship between the object distance (u), the image distance (v), and the focal length (f) of a mirror:

1/f = 1/v - 1/u

The magnification formula relates the height of the image (h') to the height of the object (h):

magnification (m) = h'/h = -v/u

Now, let's assume that the object distance (u) is twice the focal length (f), as stated in the problem. So, mathematically:

u = -2f

Substituting this value of u in the mirror formula:

1/f = 1/v - 1/(-2f)

Simplifying the equation:

1/f = 1/v + 1/2f

Considering that the focal length is positive for a concave mirror, multiplying the entire equation by 2f:

2 = 2f/v + 1

Rearranging the equation:

2f/v = 1

Now, let's substitute this relationship into the magnification formula:

m = -v/u = -v/(-2f) = v/2f

Since the height of the image (h') is directly proportional to the magnification, we can say that:

h' = mh

Substituting the value of magnification (m):

h' = (v/2f)h

Therefore, the equation shows that when the object is placed at twice the focal length from the concave mirror (u = -2f), the resulting image has a size that is half the size of the object (h' = h/2).