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 step by step, we will follow the principles of optics related to mirrors.

### Step 1: Identify the type of mirror
Given that the image is virtual and erect, we can conclude that the mirror is a **concave mirror**. Concave mirrors produce virtual images when the object is placed within the focal length.

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

### Step 3: Determine the heights of the object and the image
The height of the object (h_o) is given as 10 cm, and the height of the image (h_i) is given as 6 cm.

### Step 4: Calculate the magnification
Magnification (m) is defined as the ratio of the height of the image to the height of the object:
\[ m = \frac{h_i}{h_o} = \frac{6 \, \text{cm}}{10 \, \text{cm}} = 0.6 \]
In mirrors, magnification is also given by:
\[ m = -\frac{v}{u} \]
where \( v \) is the image distance and \( u \) is the object distance. Thus, we can write:
\[ 0.6 = -\frac{v}{u} \]
From this, we can express \( v \) in terms of \( u \):
\[ v = -0.6u \]

### Step 5: Use the mirror formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting \( f = 30 \, \text{cm} \) and \( v = -0.6u \) into the mirror formula:
\[ \frac{1}{30} = \frac{1}{-0.6u} + \frac{1}{u} \]
To combine the fractions on the right side, we find a common denominator:
\[ \frac{1}{30} = \frac{-1 + 0.6}{0.6u} = \frac{-0.4}{0.6u} \]
Now we can rewrite the equation:
\[ \frac{1}{30} = \frac{-0.4}{0.6u} \]

### Step 6: Cross-multiply to solve for \( u \)
Cross-multiplying gives:
\[ 0.6u = -0.4 \times 30 \]
Calculating the right side:
\[ 0.6u = -12 \]
Now, divide both sides by 0.6:
\[ u = \frac{-12}{0.6} = -20 \, \text{cm} \]

### Conclusion
The object is located at a distance of **-20 cm** from the mirror, indicating that it is placed in front of the mirror (the negative sign indicates the direction of the object relative to the mirror).

### Final Answer
- The type of mirror is a **concave mirror**.
- The position of the object is **-20 cm**.