a small piece of wire bent into L shape with upright and horizontal positiions of equal lengths, is placed with horizontal position along axis of concave mirrror whose radius of curvature is 10cm. If bend is 20cm from pole of mirror, then ratio of lengths of images of upright and horizontal positions of wire is___________________.

3:1

To find the ratio of lengths of the images of the upright and horizontal positions of the wire, we need to apply the mirror formula and the magnification formula for a concave mirror.

Mirror Formula:
The mirror formula relates the distance of the object (u), the distance of the image (v), and the focal length (f) of the mirror:

1/f = 1/v - 1/u

Magnification formula:
The magnification (m) of the mirror is given by the ratio of the height of the image (h') to the height of the object (h):

m = -v/u = h'/h

Given:
Radius of curvature (R) = 10 cm
Distance from the bend to the pole of the mirror (u) = 20 cm

First, let's find the focal length (f) of the mirror using the formula:

f = R/2

f = 10 cm/2
f = 5 cm

Since the focal length (f) is positive for a concave mirror, the mirror formula becomes:

1/5 = 1/v - 1/20

Now, we need to consider the upright and horizontal positions of the wire separately.

For the upright position:
The distance of the object (u1) is the distance from the pole of the mirror to the bottom end of the upright position.

u1 = 20 cm

The mirror formula becomes:

1/5 = 1/v1 - 1/20

Now we solve for v1:

1/v1 = 1/5 + 1/20
1/v1 = 4/20 + 1/20
1/v1 = 5/20
v1 = 20/5
v1 = 4 cm

For the horizontal position:
The distance of the object (u2) is the distance from the pole of the mirror to the end of the horizontal position.

u2 = 20 cm + length of the horizontal position

Given that the lengths of both the upright and horizontal positions are equal:

u2 = 20 cm + 20 cm
u2 = 40 cm

The mirror formula becomes:

1/5 = 1/v2 - 1/40

Now we solve for v2:

1/v2 = 1/5 + 1/40
1/v2 = 8/40 + 1/40
1/v2 = 9/40
v2 = 40/9 cm

Now, we can find the ratio of the lengths of the images:

Ratio = v1/v2 = (4 cm) / (40/9 cm)
Ratio = (4 cm) * (9 cm/40)
Ratio = 9/10

Therefore, the ratio of the lengths of the images of the upright and horizontal positions of the wire is 9/10.