When an object of height 1 cm its kept at a distance 4 cm from a concave mirror, its erect image of height 1.5 cm is formed at a distance 6 cm behind the mirror. Find the focal length of mirror.

Answer

To find the focal length of the concave mirror, we will use the mirror formula and the given data. Let's break down the solution step by step.

### Step 1: Identify the given values
- Height of the object (h_o) = 1 cm
- Height of the image (h_i) = 1.5 cm
- Object distance (u) = -4 cm (negative because the object is in front of the mirror)
- Image distance (v) = +6 cm (positive because the image is formed behind the mirror)

### Step 2: Use the mirror formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Where:
- f = focal length of the mirror
- u = object distance
- v = image distance

### Step 3: Substitute the values into the mirror formula
Substituting the values of u and v into the formula:
\[ \frac{1}{f} = \frac{1}{6} + \frac{1}{-4} \]

### Step 4: Calculate the right-hand side
To add the fractions, we need a common denominator. The least common multiple of 6 and 4 is 12:
\[ \frac{1}{6} = \frac{2}{12} \quad \text{and} \quad \frac{1}{-4} = \frac{-3}{12} \]
Now, substituting these values:
\[ \frac{1}{f} = \frac{2}{12} - \frac{3}{12} = \frac{-1}{12} \]

### Step 5: Solve for f
Taking the reciprocal to find f:
\[ f = -12 \text{ cm} \]

### Conclusion
The focal length of the concave mirror is:
\[ f = -12 \text{ cm} \]