Concept:

The mirror equation - If rays emanating from a point actually meet at another point after reflection and/or refraction, that point is called the image of the first point.

• The image is real if the rays actually converge to the point.
• It is virtual if the rays do not actually meet but appear to diverge from the point when produced backwards.
• \(\frac 1f = \frac 1v + \frac 1u\)
• This relation is known as the mirror equation.
• linear magnification (m) defined as the ratio of the height of the image (h') to the height of the object (h),
• \(m = \frac {h'}h = -\frac vu\)

Calculation:

ABCD is a rectangle, then AD = BC ---- (1)

Given that corners, A and B are at distances 2f and 3f from the mirror, respectively, along the principal axis, then,

u for corner A = 2f

From mirror equation,

\(\frac 1f = \frac 1v + \frac 1{2f}\)

⇒ v = 2f

Magnification  \(m = \frac {h'}h = -\frac vu = -\frac {2f}{2f} = -1\)

This means size of the image is equal to size of the object,

⇒ AD = A'D' ---- (2)

Now, u for corner B = 3f

From the mirror equation,

\(\frac 1f = \frac 1v + \frac 1{3f}\)

⇒ v = \(\frac 32\)f

Magnification, \(m = \frac {h'}h = -\frac vu = -\frac {\frac32f}{3f} = -\frac 12\)

Means size of the image is half to size of the object,

⇒ B'C' = \(\frac 12\)BC ---- (3)

From equations 1, 2 and 3, the ratio of B'C' to A'D' is,

⇒ \(\frac {B'C'}{A'D'} = \frac{\frac12BC}{AD} = \frac12\)