**Lens Formula for Concave Lens**

Consider a ray AD parallel to principal axis falling onto a concave lens of focal length f and being diverged by passing through E. The ray DE is appeared to be diverged from the focus F, so a virtual line DF is drawn that passes through the principal focus. Another ray AF passes through the optical center Intersecting the virtual ray at H forming a virtual image HI.

**From the figure,**

BC = u = object distance

IC = v = image distance = -v (for virtual image)

FC = f = focal length = -f (for concave lens)

DC = AB

Since ΔABC ~ ΔHIC

AB/HI = BC/IC

AB/HI = u/-v ………… (i)

Similarly, in ΔHIC and ΔDCF

DC/HI = FC/IF

AB/HI = FC/(FC – IC)

AB/HI = -f/{-f -(-v)}

AB/HI = -f/(-f + v) = f/(f-v) ……… (ii)

Equating equation (i) and (ii)

u/-v = f/(f-v)

uf - uv = -vf

uf + vf = uv

dividing both sides by uvf, we get

1/f = 1/u + 1/v

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