The image formed by a concave mirror is twice the size of the object. The focal length of the mirror is 20 cm. The distance of the object from the mirror `is//are`

Options

A. `10 cm`
B. `30 cm`
C. `25 cm`
D. `15 cm`

Explanation

To solve the problem step by step, we will use the mirror formula and the magnification concept for concave mirrors.

### Step 1: Understand the given information
- The image formed by the concave mirror is twice the size of the object (magnification, m = 2).
- The focal length of the concave mirror, F = -20 cm (negative because it is a concave mirror).

### Step 2: Write the magnification formula
The magnification (m) for mirrors is given by:
\[ m = -\frac{V}{U} \]
Where:
- V = image distance from the mirror
- U = object distance from the mirror

Since the image is twice the size of the object, we have:
\[ m = 2 \]
Thus,
\[ 2 = -\frac{V}{U} \]
From this, we can express V in terms of U:
\[ V = -2U \]

### Step 3: Use the mirror formula
The mirror formula is given by:
\[ \frac{1}{F} = \frac{1}{V} + \frac{1}{U} \]

Substituting the known values:
- F = -20 cm
- V = -2U

We can substitute V into the mirror formula:
\[ \frac{1}{-20} = \frac{1}{-2U} + \frac{1}{U} \]

### Step 4: Simplify the equation
Now, we simplify the equation:
\[ \frac{1}{-20} = -\frac{1}{2U} + \frac{1}{U} \]

Finding a common denominator for the right side:
\[ \frac{1}{-20} = -\frac{1}{2U} + \frac{2}{2U} \]
\[ \frac{1}{-20} = \frac{1}{2U} \]

### Step 5: Cross-multiply to solve for U
Cross-multiplying gives:
\[ 1 \cdot 2U = -20 \cdot 1 \]
\[ 2U = -20 \]
\[ U = -10 \text{ cm} \]

### Step 6: Find the image distance V
Now substituting U back to find V:
\[ V = -2U = -2(-10) = 20 \text{ cm} \]

### Step 7: Determine the distances
Since we have two possible cases for the object distance:
1. **U = -10 cm** (indicating a virtual image)
2. **U = -30 cm** (indicating a real image)

### Conclusion
The distances of the object from the mirror can be:
- U = -10 cm (virtual image)
- U = -30 cm (real image)

Thus, the distances of the object from the mirror are **10 cm and 30 cm**.

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