A candle flame 0.5 cm high is kept between a wall and a concave mirror of focal length 1.5 m such that its image 1.5 cm high is formed on the wall. Find the distance of candle from the wall and also the distance between the wall and the mirror.

Options

A. 4 m, 6 m
B. 6 m, 7 m
C. 6 m, 8 m
D. 9 m, 7 m

Answer

To solve the problem, we need to find the distance of the candle from the wall and the distance between the wall and the concave mirror.

### Step 1: Understand the given data
- Height of the candle (object), \( h_o = 0.5 \, \text{cm} \)
- Height of the image, \( h_i = 1.5 \, \text{cm} \)
- Focal length of the concave mirror, \( f = -1.5 \, \text{m} \) (negative because it is a concave mirror)

### Step 2: Use the magnification formula
The magnification \( m \) is given by the formula:
\[ m = \frac{h_i}{h_o} = -\frac{b}{u} \]
Where:
- \( b \) is the image distance
- \( u \) is the object distance

Substituting the values:
\[ m = \frac{1.5}{0.5} = 3 \]
Thus,
\[ -\frac{b}{u} = 3 \implies b = -3u \]

### Step 3: Use the mirror formula
The mirror formula is:
\[ \frac{1}{f} = \frac{1}{b} + \frac{1}{u} \]
Substituting \( b = -3u \) into the mirror formula:
\[ \frac{1}{-1.5} = \frac{1}{-3u} + \frac{1}{u} \]

### Step 4: Simplify the equation
Rearranging the equation:
\[ \frac{1}{-1.5} = -\frac{1}{3u} + \frac{1}{u} \]
Finding a common denominator for the right side:
\[ \frac{1}{-1.5} = \frac{-1 + 3}{3u} = \frac{2}{3u} \]
Thus, we have:
\[ \frac{1}{-1.5} = \frac{2}{3u} \]

### Step 5: Cross-multiply to solve for \( u \)
Cross-multiplying gives:
\[ -1.5 \cdot 2 = 3u \]
\[ -3 = 3u \implies u = -1 \, \text{m} \]

### Step 6: Find the distance of the candle from the wall
Since \( u \) is negative, it indicates that the object (candle) is 1 meter in front of the mirror. The distance of the candle from the wall is:
\[ \text{Distance from wall} = \text{Distance from mirror} = 1 \, \text{m} \]

### Step 7: Find the distance between the wall and the mirror
The distance between the wall and the mirror is:
\[ \text{Distance between wall and mirror} = |u| + |b| \]
Where \( b \) can be calculated as:
\[ b = -3u = -3(-1) = 3 \, \text{m} \]
Thus, the distance between the wall and the mirror is:
\[ \text{Distance} = 1 + 3 = 4 \, \text{m} \]

### Final Results
- Distance of the candle from the wall: **1 m**
- Distance between the wall and the mirror: **4 m**