How far should an object be placed from the pole of a converging mirror of focal length 20 cm to form a real image of the size exactly $\frac{1}{4}$th the size of the object ?

Focal length, f = -20 cm

Let object distance be u and image distance be v.

Magnification is, m = -v/u = -1/4 (for real image m is negative)

Therefore, v = u/4

Now,

1/f = 1/v + 1/u

=> -1/20 = 4/u + 1/u

=> -1/20 = 5/u

=> u = -100 cm

Thus, the object must be placed at a distance 100 cm from the pole of the concave mirror.