Practice question

**Combination of lenses in contact** effective power P = P₁+P₂, effective focal length 1/F = 1/f₁ + 1/f₂, for thin lenses in contact. If separated by d, 1/F =1/f₁+1/f₂ - d/(f₁ f₂). Power adds algebraically, converging + with diverging - can cancel, used to correct aberrations and design achromatic doublets. Focal length: f = (R/2) = (-30/2) = -15 cm (concave mirror). Object distance: u = -45 cm . Mirror equation: (1/v) + (1/u) = (1/f) . (1/v) + (1/-45) = (1/-15) ⇒ (1/v) = (1/-15) + (1/45) = (-3 + 1/45) = (-2/45) . v = -(45/2) = -22.5 cm (real image). Substituting values