Derive th lens maker's formula.

(a) A point object is placed in front of a double convex lens ( of refractive index n = n_(2) // n_(1) with respect to air ) with its spherical faces of radii of curvature R_(1) and R_(2) . Show the path of rays due to refraction at first and subsequently at the second surface to obtain the formation of the real image of the object. Hence obtain the lens-maker's formula for a thin lens. A double convex lens having both faces of the same radius of curvature has refractive index 1.55. Find out the radius of curvature of the lens required to get the focal length of 20cm.

Draw a ray diagram showing the formation of the image by a point object on the principal axis of a spherical convex surface separating two media of refractive indices n_(1) and n_(2) , when a point source is kept in rarer medium of refractive index n_(1) . Derive the relation between object and image distance in terms of refractive index of the medium and radium of curvature of the surface. Hence obtain the expression for lens-maker's formula in the case of thin convex lens.

Derive the quadratic formula from the standard form (ax^(2)+bx+c=0) of a quadratic equation.

Derive the section formula for the internal division in three dimensions.