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 2*f*.

Using equation (1), we get:

∴
is negative, i.e., *v*
is negative.

Therefore,
the image lies beyond 2*f*.

**(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.

**
**