A beam of light converges towards a point O, behind a convex mirror of focal length 20cm.
Q. Similarly, as in above question when point O is 30cm behind the mirror.

Options

A. `2`(virtual, inverted)
B. `3`(real, inverted)
C. `3`(virtual, enlarged)
D. `1(`real, enlarged)

Explanation

To solve the problem step by step, we will follow the concepts of geometrical optics, specifically focusing on the convex mirror and its properties.

### Step 1: Identify the Given Values
- Focal length of the convex mirror (f) = +20 cm (positive because it is a convex mirror)
- Distance of the object (u) = -30 cm (negative because the object is behind the mirror)

### Step 2: Use the Mirror Formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Where:
- \( f \) = focal length
- \( v \) = image distance
- \( u \) = object distance

### Step 3: Substitute the Values into the Mirror Formula
Substituting the values of \( f \) and \( u \) into the mirror formula:
\[ \frac{1}{20} = \frac{1}{v} + \frac{1}{-30} \]

### Step 4: Rearrange the Equation
Rearranging the equation gives:
\[ \frac{1}{v} = \frac{1}{20} + \frac{1}{30} \]

### Step 5: Find a Common Denominator
The common denominator for 20 and 30 is 60. Thus, we can rewrite the fractions:
\[ \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{30} = \frac{2}{60} \]
So,
\[ \frac{1}{v} = \frac{3}{60} + \frac{2}{60} = \frac{5}{60} \]

### Step 6: Calculate the Image Distance (v)
Now, taking the reciprocal to find \( v \):
\[ v = \frac{60}{5} = 12 \text{ cm} \]
(Note: The positive sign indicates that the image is virtual and located behind the mirror.)

### Step 7: Calculate the Magnification (M)
The magnification (M) is given by the formula:
\[ M = -\frac{v}{u} \]
Substituting the values of \( v \) and \( u \):
\[ M = -\frac{12}{-30} = \frac{12}{30} = \frac{2}{5} \]

### Step 8: Interpret the Results
The magnification of \( \frac{2}{5} \) means that the image is virtual, upright, and smaller than the object.

### Final Results
- Image distance (v) = 12 cm (behind the mirror)
- Magnification (M) = \( \frac{2}{5} \)