A candle flame of 3 cm is placed at 300 cm from a wall. A concave mirror is kept at distance x from the wall in such a way that image of the flame on the wall is 9 cm. Then x is -

Options

A. 339 cm
B. 900 cm
C. 450 cm
D. 423 cm

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the problem
We have a candle flame of height 3 cm placed at a distance of 300 cm from a wall. A concave mirror is placed at a distance \( x \) from the wall, and the image of the candle flame on the wall is 9 cm tall. We need to find the value of \( x \).

### Step 2: Set up the parameters
- Height of the object (candle flame), \( h_o = 3 \) cm (positive since it is above the principal axis).
- Height of the image, \( h_i = -9 \) cm (negative since the image is inverted).
- Distance from the candle to the wall, \( d = 300 \) cm.

### Step 3: Use the magnification formula
The magnification \( m \) can be defined in two ways:
1. \( m = \frac{h_i}{h_o} \)
2. \( m = -\frac{v}{u} \)

Where:
- \( v \) is the image distance from the mirror,
- \( u \) is the object distance from the mirror.

### Step 4: Calculate magnification
Using the first formula:
\[ m = \frac{-9}{3} = -3 \]

### Step 5: Express \( u \) and \( v \)
The object distance \( u \) can be expressed as:
\[ u = x - 300 \quad (\text{since the object is 300 cm from the wall and the mirror is at distance } x) \]

The image distance \( v \) is:
\[ v = x \quad (\text{since the image is formed on the wall, which is at distance } x \text{ from the mirror}) \]

### Step 6: Set up the equation using magnification
Using the second formula for magnification:
\[ -3 = -\frac{v}{u} \]
Substituting \( v \) and \( u \):
\[ -3 = -\frac{x}{x - 300} \]

### Step 7: Simplify the equation
Removing the negative signs:
\[ 3 = \frac{x}{x - 300} \]

Cross-multiplying gives:
\[ 3(x - 300) = x \]
\[ 3x - 900 = x \]

### Step 8: Solve for \( x \)
Rearranging the equation:
\[ 3x - x = 900 \]
\[ 2x = 900 \]
\[ x = \frac{900}{2} = 450 \text{ cm} \]

### Conclusion
The distance \( x \) from the wall to the concave mirror is **450 cm**.

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