The image of the flame of a candle, in a concave mirror, is formed at a distance of 30 cm in front of the mirror, Height of the flame is 10 cm and its image is 5 cm high. Distance of the focus from mirror is

Options

A. 10 cm
B. 15 cm
C. 20 cm
D. 30 cm

Explanation

To solve the problem step by step, we will use the concepts of magnification and the mirror formula.

### Step 1: Understand the given data
- The distance of the image (V) from the mirror = -30 cm (negative because the image is in front of the mirror)
- The height of the object (O) = 10 cm
- The height of the image (I) = -5 cm (negative because the image is inverted)

### Step 2: Calculate the magnification (m)
The magnification (m) is given by the formula:
\[ m = \frac{h_i}{h_o} = \frac{-V}{U} \]
Where:
- \( h_i \) = height of the image
- \( h_o \) = height of the object
- \( V \) = image distance
- \( U \) = object distance (which we need to find)

From the given data:
\[ m = \frac{-5}{10} = -0.5 \]

### Step 3: Relate magnification to distances
Using the magnification formula:
\[ m = \frac{-V}{U} \]
Substituting the known values:
\[ -0.5 = \frac{-(-30)}{U} \]
This simplifies to:
\[ -0.5 = \frac{30}{U} \]

### Step 4: Solve for U
Cross-multiplying gives:
\[ U = \frac{30}{-0.5} = -60 \text{ cm} \]

### Step 5: Use the mirror formula
The mirror formula is:
\[ \frac{1}{F} = \frac{1}{U} + \frac{1}{V} \]
Substituting the values of U and V:
\[ \frac{1}{F} = \frac{1}{-60} + \frac{1}{-30} \]

### Step 6: Calculate the right-hand side
Finding a common denominator (which is 60):
\[ \frac{1}{F} = \frac{-1}{60} + \frac{-2}{60} = \frac{-3}{60} \]

### Step 7: Solve for F
Now, we can find F:
\[ F = \frac{60}{-3} = -20 \text{ cm} \]

### Conclusion
The distance of the focus from the mirror is **-20 cm**.

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