According to the new Cartesian system the Pole (P) of the mirror is taken as the origin and the Principal axis is taken as the horizontal X- axis. The sign conventions for it are as follows:
The object is always placed on the left side of the mirror.
All the distances measured to the right side of the origin are taken as positive, while the distances measured to the left side of the origin are taken as negative.
All the distances parallel to the principal axis of the mirror are measured from the pole of the mirror.
Distances measured vertically upwards from the principal axis of the mirror are taken as positive.
Distances measured vertically downwards from the principal axis of the mirror are taken as negative.
The focal length of Concave mirror is negative while that of convex mirror is positive.
The figure below illustrates the conventions used.
Given;
Magnification (m) = \(\frac13\) (virtual image is always erect).
Object distance (u) = -18cm.
By Magnification Formula;
Magnification, m =\(-\frac{v}u\)
Where,
v is the image distance
u is the object distance
⇒ \(\frac13\) = \(-\frac{v}{-18}\)
⇒ v = \(\frac{18}{3}\) = 6
⇒ v = 6cm.
∴ Image distance (v) = 6cm.
By mirror formula;
\(\frac1v+\frac1u=\frac1f\)
⇒ \(\frac16+\frac1{-18}=\frac1f\)
⇒ \(\frac16-\frac1{18}=\frac1f\)
⇒ \(\frac1f=\frac{3-1}{18}=\frac2{18}\)
⇒ f = \(\frac{18}{2}\) = 9 cm
Therefore, the focal length is 9 cm. Since the focal length is positive, therefore the mirror is convex mirror.
Figure below shows the ray diagram for the same.
