A concave mirror gives an image three times as large as the object placed at a distance of 20 cm from it. For the image to be real, the focal length should be

Options

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

Explanation

To find the focal length of a concave mirror that produces an image three times larger than the object, we can follow these steps:

### Step 1: Identify the given values
- The magnification (m) is given as 3 (since the image is real, we take it as -3).
- The object distance (u) is given as 20 cm (which is negative for a concave mirror, so u = -20 cm).

### Step 2: Write the magnification formula
The magnification (m) for a mirror is given by the formula:
\[ m = \frac{-v}{u} \]
where \( v \) is the image distance.

### Step 3: Substitute the values into the magnification formula
Since we know that the magnification is -3 and the object distance \( u = -20 \) cm, we can substitute these values into the magnification formula:
\[ -3 = \frac{-v}{-20} \]

### Step 4: Solve for image distance (v)
Rearranging the equation gives:
\[ -3 = \frac{v}{20} \]
Multiplying both sides by 20:
\[ v = -60 \text{ cm} \]

### Step 5: Use the mirror formula to find the focal length (f)
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]
Substituting the values we have:
\[ \frac{1}{f} = \frac{1}{-20} + \frac{1}{-60} \]

### Step 6: Calculate the right-hand side
Finding a common denominator (which is 60):
\[ \frac{1}{f} = \frac{-3}{60} + \frac{-1}{60} \]
\[ \frac{1}{f} = \frac{-4}{60} \]
\[ \frac{1}{f} = \frac{-1}{15} \]

### Step 7: Solve for focal length (f)
Taking the reciprocal gives:
\[ f = -15 \text{ cm} \]

### Conclusion
The focal length of the concave mirror is -15 cm.