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When an object is placed at a distance of 50 cm from a concave mirror , the magnification produced is - 1/2 . where should the object be placed to get a magnification of -1/5

Let, focal length be f.

Now,

Object distance, u = -50 cm

Magnification, m = -1/2

Image distance = v

So, -v/u = -1/2

=> v = u/2 = -25 cm

So,

1/f = 1/v + 1/u

=> 1/f = -1/25 – 1/50

=> f = -50/3 cm

Now,

Magnification is, m = -1/5

So, -v/u = -1/5

=> v = u/5

Thus,

1/f = 1/v + 1/u

=> -3/50 = 5/u + 1/u

=> -3/50 = 6/u

=> u = -100 cm

So, the object must be placed at a distance 100 cm from the mirror.

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Here,

m = -1/2 (since the images is real)

u = - 50 cm.

We know,

m = -v/u

=> -1/2 = -v / -50 => v = -50/2 = -25 cm

Also, according to the mirror formula,

1/f = 1/v + 1/u

=> 1/f = -1/25 - 1/50

=> 1/f = (-2-1)/50

=> 1/f = -3/50 => f = -50/3 cm

So, the focal length of mirror is 60 cm.

Now, we are given that m = -1/5

m = -v/u

=> -1/5 = -v/u => v = u/5

According to mirror formula,

1/f = 1/u + 1/v

=> -3/50 = 1/u + 5/u

=> -3/50 = 6/u

Therefore, u = -300/3 = -100 cm

Therefore, required distance of the object to produce a magnification of -1/5 is 100 cm.

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