An image to be two third size of the object is formed by concave mirror the distance of 20 cm from mid find the focal length of the mirror?

Question

Answer

Given:
- The image formed by a concave mirror is two-thirds the size of the object.
- The distance between the mirror and the midpoint of the object is 20 cm.

To find:
- The focal length of the mirror.

Formula:
The magnification (m) of an image formed by a concave mirror is given by:
m = -(v/u)

Where:
m = Magnification
v = Image distance from the mirror
u = Object distance from the mirror

Step 1: Identify the given values
- Object distance (u) = 20 cm
- Magnification (m) = 2/3

Step 2: Calculate the image distance (v)
Using the magnification formula, we can rearrange it to find v:
v = -(m * u)

Substituting the given values:
v = -(2/3 * 20)
v = -40/3 cm

Step 3: Determine the focal length (f)
The focal length (f) of a concave mirror can be calculated using the mirror formula:
1/f = 1/v - 1/u

Substituting the values:
1/f = 1/(-40/3) - 1/20

To simplify the equation, we take the reciprocal of both sides:
f = 1/((1/(-40/3)) - (1/20))
f = 1/((-3/40) - (1/20))

To add the fractions, we need a common denominator:
f = 1/((-3/40) - (2/40))
f = 1/(-5/40)
f = -40/5
f = -8 cm

Step 4: Interpret the result
The focal length of the concave mirror is -8 cm. The negative sign indicates that the mirror is a concave mirror, as the focal length is measured on the same side as the object.

Therefore, the focal length of the concave mirror is -8 cm.

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