A concave mirrorr of focal length `15 cm` forms an image having twice the linear dimensions of the object. The position of the object when the image is virtual will be

Options

A. 45 cm
B. 30 cm
C. `7.5 cm`
D. `22.5 cm`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the given values
We have a concave mirror with a focal length (f) of -15 cm (the negative sign indicates that it is a concave mirror). The magnification (M) is given as +2, indicating that the image is virtual and upright.

### Step 2: Use the magnification formula
The magnification (M) is defined as:
\[ M = -\frac{V}{U} \]
Where:
- V is the image distance
- U is the object distance

Given that M = +2, we can set up the equation:
\[ 2 = -\frac{V}{U} \]
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 values we have:
\[ \frac{1}{-15} = \frac{1}{-2U} + \frac{1}{U} \]

### Step 4: Simplify the equation
To simplify, we can find a common denominator for the right side:
\[ \frac{1}{-15} = \frac{-1 + 2}{2U} \]
This simplifies to:
\[ \frac{1}{-15} = \frac{1}{2U} \]

### Step 5: Cross-multiply to solve for U
Cross-multiplying gives us:
\[ 1 \cdot 2U = -15 \cdot 1 \]
So:
\[ 2U = -15 \]
Dividing both sides by 2 gives:
\[ U = -7.5 \, \text{cm} \]

### Step 6: Interpret the result
The object distance U is -7.5 cm. The negative sign indicates that the object is located on the same side as the incoming light (which is typical for real objects in mirror problems).

### Final Answer
The position of the object when the image is virtual is **7.5 cm** (considering the absolute value).

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