What is the relation between focal length and radius of curvature of a concave mirror what is radius of curvature?

An object 5 cm in length is held 25 cm away from a converging lens of focal length 10 cm. Draw the ray diagram and find the position, size and the nature of the image formed.

Converging lens means a convex lens. As the distances given in the question are large, so we choose a scale of 1: 5, i.e., 1 cm represents 5 cm. Therefore, on this scale 5 cm high object, object distance of 25 cm and focal length of 10 cm can be represented by 1 cm high, 5 cm and 2 cm lines respectively. Now, we draw the ray diagram as follows:(i) Draw a horizontal line to represent the principal axis of the convex lens.(ii) Centre line is shown by DE.(iii) Mark two foci F and F' on two sides of the lens, each at a distance of 2 cm from the lens.(iv) Draw an arrow AB of height 1 cm on the left side of lens at a distance of 5 cm from the lens.(v) Draw a line AD parallel to principal axis and then, allow it to pass straight through the focus (F') on the right side of the lens.(vi) Draw a line from A to C (centre of the lens), which goes straight without deviation.(vii) Let the two lines starting from A meet at A'.(viii) Draw AB', perpendicular to the principal axis from A'.(ix) Now AB', represents the real, but inverted image of the object AB.(x) Then, measure CB' and A'B'. It is found that CB' = 3.3 cm and A'B' = 0.7 cm.

(xi) Thus the final position, nature and size of the image A'B' are: (a) Position of image A'B' = 3.3 cm × 5 = 16.5 cm from the lens on opposite side. (b) Nature of image A’B’: Real and inverted.

The focal length of a spherical mirror is equal to half of its radius of curvature
{f = 1/2 R}

Proof: In below figure1 and 2, a ray of light BP' travelling parallel to the principal axis PC is incident on a spherical mirror PP'. After reflection, it goes along P'R, through the focus F P is the pole and F is the focus of the mirror. The distance PF is equal to the focal length f. C is the centre of curvature. The distance PC is equal to the radius of curvature R of the mirror. P'C is the normal to the mirror at the point of incidence P'

For a concave mirror:
In figure,

∠BP'C	=	∠P'CF (alternate angles)
and ∠BP'C	=	∠P'F (law of reflection, ∠i = ∠r)
Hence ∠P'CF	=	∠CP'F
∴ FP'C is isosceles.
Hence, P'F	=	FC

If the aperture of the mirror is small, the point P' is very close to the point P,

then P'F	=	PF
∴PF	=	FC
	=	1/2 PC
or f	=	1/2 R

For a convex mirror:
In figure,

∠BP'N	=	FC∠P' (corresponding angles)
∠BP'N	=	∠NP'R (law of reflection, ∠i = ∠r)
and ∠NP'R	=	∠CP'F (vertically opposite angles)
Hence ∠FCP'	=	∠CP'F
∴ FP'C is isosceles.
Hence, P'F	=	FC