Question

The distance between two points sources of light is 24 cm find out where you should place a converging lens of focal length 9cm between the sources so that image of both the sources are formed at the same point

Answer 1

Let lens is placed x distance from first source(left side)

then object distance, u = -x cm

focal length of lens , f = +9cm [ lens is covering , means lens is convex lens ]

now, using lens maker formula,

1/v - 1/u = 1/f

or, 1/v - 1/-x = 1/9

or, 1/v = 1/9 - 1/x

or, 1/v = (x - 9)/9x

v = 9x/(x - 9) ........(1)

lens is located (24 - x)cm from 2nd source (right side)

then object distance, u' =(24 - x)cm

focal length , f = -9cm

[ here we take u positive and f negative because source is placed right side of lens ]

from lens maker formula,

1/v - 1/u = 1/f

or, 1/v - 1/(24 - x) = 1/-9

or, 1/v = 1/(24 - x) - 1/9

or, 1/v = (9 - 24 + x)/9(24 - x)

or, v = 9(24 - x)/(x - 15) ............(2)

from equations (1) and (2),

9x/(x - 9) = 9(24 - x)/(x - 15)

or, x(x - 15) = (x - 9)(24 - x)

or, x² - 15x = 24x - x² - 216 + 9x

or, 2x² - 48x + 216 = 0

or, x² - 24x + 108 = 0

or, x = 6 and 18

hence, lens is placed 6cm from first source and 18cm from 2nd source.

Answer 2

Answer:

Let the first point source be located at the distance u1 & its image is formed at the distance v1. Similarly, the second point source be located at the distance u2 & its image located at v2.

Given: focal length f = 9 cm

Distance between two sources = 24 cm

For first point source:

As both the images are at the same point therefore v1 = v2 = v

..(1)

For second point source:

…(2)

[u1 + u2 = 24 cm]

Adding (1) & (2)

24 u1 – u12 = 108

U12-24u1 + 108 = 0

U12 - 18u1 - 6u1 + 108 = 0

(u1-6)(u1-18) = 0

u1 = 6 cm or u1 = 18 cm

So the lens is kept at the distance of 6 cm or 18 cm from the first point source.