Radii of curvature of a concavo-convex lens (refractive index = 1.5) are 40 cm and 20 cm as shown. The convex side is silvered. The distance x (in cm) on the principal axis where an object is placed so that its image is created on the object itself, is given as $4 β$ . Find the value of $β$ .

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F l e n s ​ 1 ​ = ( 1.5 − 1 ) [ − 40 1 ​ − − 20 1 ​ ] = 80 1 ​ ⇒ F ℓ ​ = 80 cm F m ​ = − 2 20 ​ = − 10 cm F e q ​ 1 ​ = f m ​ 1 ​ − f ℓ ​ 2 ​ = − 10 1 ​ − 80 2 ​ ⇒ f e q ​ = − 8 cm Object must be at the centre of curvature: x = 16 cm = 4 β ⇒ β = 4 .

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