Use the mirror equation to deduce that:

(a) an object placed between f and 2 f of a concave mirror produces a real image beyond 2 f.

(b) a convex mirror always produces a virtual image independent of the location of the object.

(c) the virtual image produced by a convex mirror is always diminished in size and is located between the focus and the pole.

(d) an object placed between the pole and focus of a concave mirror produces a virtual and enlarged image.

[ Note: This exercise helps you deduce algebraically properties of

images that one obtains from explicit ray diagrams.]

(a) For a concave mirror, the focal length (f) is negative.

∴f < 0

When the object is placed on the left side of the mirror, the object distance (u) is negative.

∴u < 0

For image distance v, we can write the lens formula as:

The object lies between f and 2f.

Using equation (1), we get:

∴ is negative, i.e., v is negative.

Therefore, the image lies beyond 2f.

(b) For a convex mirror, the focal length (f) is positive.

∴ f > 0

When the object is placed on the left side of the mirror, the object distance (u) is negative.

∴ u < 0

For image distance v, we have the mirror formula:

Thus, the image is formed on the back side of the mirror.

Hence, a convex mirror always produces a virtual image, regardless of the object distance.

(c) For a convex mirror, the focal length (f) is positive.

∴f > 0

When the object is placed on the left side of the mirror, the object distance (u) is negative,

∴u < 0

For image distance v, we have the mirror formula:

Hence, the image formed is diminished and is located between the focus (f) and the pole.

(d) For a concave mirror, the focal length (f) is negative.

∴f < 0

When the object is placed on the left side of the mirror, the object distance (u) is negative.

∴u < 0

It is placed between the focus (f) and the pole.

For image distance v, we have the mirror formula:

The image is formed on the right side of the mirror. Hence, it is a virtual image.

For u < 0 and v > 0, we can write:

Magnification, m > 1

Hence, the formed image is enlarged.