An object placed at 20cm in front of a concave mirror produces three times magnified real image. What is the focal length of the concave mirror?

Options

A. 15cm
B. 6.6cm
C. 10cm
D. 7.5cm

Explanation

To find the focal length of the concave mirror given that an object placed at 20 cm in front of it produces a three times magnified real image, we can follow these steps:

### Step 1: Understand the given data
- Object distance (u) = -20 cm (the negative sign indicates that the object is in front of the mirror)
- Magnification (m) = -3 (the negative sign indicates that the image is real and inverted)

### Step 2: Use the magnification formula
The magnification (m) for mirrors is given by the formula:
\[ m = \frac{h'}{h} = \frac{-v}{u} \]
Where:
- \( h' \) = height of the image
- \( h \) = height of the object
- \( v \) = image distance
- \( u \) = object distance

From the magnification formula, we can rearrange it to find the image distance \( v \):
\[ -3 = \frac{-v}{-20} \]
This simplifies to:
\[ v = 3 \times 20 = 60 \text{ cm} \]

### Step 3: Use the mirror formula
The mirror formula relates the object distance (u), image distance (v), and focal length (f) of the mirror:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]

### Step 4: Substitute the values into the mirror formula
Substituting the values we have:
- \( v = 60 \) cm
- \( u = -20 \) cm

So,
\[ \frac{1}{f} = \frac{1}{60} + \frac{1}{-20} \]

### Step 5: Calculate the right side of the equation
Calculating the right side:
\[ \frac{1}{f} = \frac{1}{60} - \frac{3}{60} = \frac{1 - 3}{60} = \frac{-2}{60} = -\frac{1}{30} \]

### Step 6: Find the focal length
Now, taking the reciprocal to find \( f \):
\[ f = -30 \text{ cm} \]

### Conclusion
The focal length of the concave mirror is \( -30 \) cm.

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