When the distance of an object from a concave mirror is decreased from 15 cm to 9cm . the image gets magnified 3 times than that is first case. calculate the focal length of the mirror
Answers
To answer this question we need to know the formula for magnification.
Magnification = Image distance / Object distance
In this case :
Let the magnification in the first case be x.
The magnification in the second case will be 3x.
The mirror formula is :
1/u + 1/v = 1/f
The first case
Magnification = x
Object distance = 25 cm
Image distance = Object distance × Magnification
= 15x cm
Now :
1/15 + 1/15 x = 1/f
Lcm = 15x
Solving we have :
(1 + x)/15x = 1/f
f = 15x/(1 + x)
Second case
Object distance = 9 cm
magnification = 3x
Image distance = 27x cm
1/9 + 1/27x = 1/f
Lcm = 27x
(3x + 1)/27x = 1/f
f = 27x/(3x + 1)
Since we have two values of f we can equate them as follows:
27x/(3x + 1) = 15x/(x + 1)
Solving for x we have :
27x(x + 1) = 15x(3x + 1)
27x² + 27x = 45x² + 15x
Collecting like terms together
45x² - 27x² = 27x - 15x
18x² = 12x
Divide through by x
18x = 12
x = 12/18
x = 2/3
Substitute to get f:
f = (27 × 2/3)/(3 × 2/3 + 1)
f = 18/3