Newton’s equation:
Let us consider an object AB is placed at a distance u from the pole of a concave mirror of focal length f. The image A/B/ is formed at a distance of v from the pole. Let us consider the distance between object and focus is FA = x = (u – f) and distance between image and focus is FA/ = y = (v – f). Using mirror formula we get, 
Or, 
Or, uf + vf = uv
Or, uv – uf – vf = 0
Or, uv – uf – vf + 
=
Or, u(v – f) – f(v – f) =
Or, (u – f)(v – f) =
xy =
This is known as Newton’s equation.
Graph:
The mirror formula is 
.
Let us consider 
= y, 
= x and 
= c
Then y + x = c
Or, y = – x + c
This is equation of straight line.
If we plot v along y axis and u along x axis respectively then the graph is hyperbola. When the object is at infinity (u = 
) then the image is at focus (v = f) and for u = f, v = . When u = v, a straight line cut the hyperbola at (2f, 2f).
Paraxial rays, marginal rays and spherical aberration:
Paraxial rays: Those rays which incident on the spherical mirror near to the principle axis of the mirror are called paraxial ray.
Marginal Ray: Those rays incident on the mirror just closer to the edge of the mirror is called marginal ray.
For a spherical mirror with small aperture, the marginal rays are converge at a point on the principal axis is called marginal focus. Similarly paraxial rays also converge at a point on the principal axis is called paraxial focus.
The focal length of a spherical mirror due to marginal ray are less than that of paraxial ray.
The failure of the paraxial and the marginal rays to pass through the same point after reflection through a spherical surface is called spherical aberration.
Correction of spherical aberration:
Spherical aberration can be eliminated if a concave mirror is replaced by a parabolic mirror. It has the ability to focus the marginal and paraxial rays at a point on the principal axis. Therefore, the intensity of the reflected beam of light for parabolic mirror is more. For that reason, it is used in search light, motor headlight reflector. It is also used in radio telescope.
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