An object of length 10 cm is placed at right angles to the principal axis of a mirror of radius of curvature 60 cm such that its image is virtual, erect and has a length 6 cm. What kind of mirror it is and also determine the position of the object ?

Options

A. -20 cm
B. 20 cm
C. -30 cm
D. 30 cm

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the type of mirror
Given that the image is virtual, erect, and diminished, we can conclude that the mirror is a **convex mirror**. In convex mirrors, images formed are always virtual, erect, and smaller than the object.

### Step 2: Determine the focal length of the mirror
The radius of curvature (R) of the mirror is given as 60 cm. The focal length (f) of a mirror is related to its radius of curvature by the formula:
\[ f = \frac{R}{2} \]
Substituting the value of R:
\[ f = \frac{60 \, \text{cm}}{2} = 30 \, \text{cm} \]
Since it is a convex mirror, the focal length is positive:
\[ f = +30 \, \text{cm} \]

### Step 3: Calculate the magnification
The magnification (m) is given by the ratio of the height of the image (h_i) to the height of the object (h_o):
\[ m = \frac{h_i}{h_o} = \frac{6 \, \text{cm}}{10 \, \text{cm}} = 0.6 \]

### Step 4: Relate magnification to object distance
The magnification can also be expressed in terms of object distance (u) and image distance (v):
\[ m = -\frac{v}{u} \]
Since we already know m = 0.6, we can write:
\[ 0.6 = -\frac{v}{u} \quad \Rightarrow \quad v = -0.6u \]

### Step 5: Use the mirror formula
The mirror formula relates object distance (u), image distance (v), and focal length (f):
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the values we have:
\[ \frac{1}{30} = \frac{1}{-0.6u} + \frac{1}{u} \]

### Step 6: Solve for u
To solve for u, we first find a common denominator:
\[ \frac{1}{30} = \frac{-1 + 0.6}{0.6u} = \frac{-0.4}{0.6u} \]
Cross-multiplying gives:
\[ -0.4 = \frac{30 \times 0.6u}{1} \quad \Rightarrow \quad -0.4 = 18u \]
Thus:
\[ u = \frac{-0.4}{18} = -\frac{2}{90} = -\frac{1}{45} \, \text{m} = -20 \, \text{cm} \]

### Conclusion
The position of the object is at **-20 cm** from the mirror, indicating that it is located on the same side as the object (which is typical for virtual images in convex mirrors).

### Summary
- The type of mirror is **convex**.
- The position of the object is **-20 cm**.

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