A concave spherical mirror forms an inverted image different in size from the object by a factor a > 1. The distance between object and image is d. Find the focal length of the mirror. (Use any variable or symbol stated above as necessary.)

(b) Suppose the mirror is convex, an upright image is formed, and a < 1. Determine the focal length of the mirror. (Use any variable or symbol stated above as necessary.)

To find the focal length of a concave spherical mirror in this scenario, we need to use the mirror formula:

1/f = 1/di + 1/do

Where:
- f is the focal length of the mirror
- di is the image distance from the mirror
- do is the object distance from the mirror

From the given information, we know that the image is inverted and different in size (a > 1). This means that di is negative. Also, the distance between the object and image is d, so the object distance is -d.

Substituting these values into the mirror formula:

1/f = 1/di + 1/do
= 1/(-d) + 1/(-d - di)
= -1/d + -1/(d + di)

Using the fact that a = di/do, we can substitute di = a * do into the equation:

1/f = -1/d + -1/(d + a * do)

Now, let's move all the terms to the left side:

1/f + 1/d - 1/(d + a * do) = 0

Combining the terms:

(d + a * do - f) / (f * d * (d + a * do)) = 0

Since this is an equation, the numerator must be zero:

d + a * do - f = 0

Now, we can solve this equation to find the focal length (f):

f = d + a * do

So, the focal length of the concave spherical mirror is given by f = d + a * do.

For the convex mirror scenario, where an upright image is formed and a < 1, the same equation can be used:

f = d + a * do

Since a < 1 in this case, the focal length will be a positive value.

f = pq/(p+q) (you use d, I use q)

so obviously there is not enough info given.