The focal length of a concave mirror is 50cm. Where an object be placed so that its image is two times and inverted

Options

A. 75 cm
B. 72 cm
C. 63 cm
D. 50 cm

Answer

To solve the problem of finding the object distance for a concave mirror where the image is twice the size of the object and inverted, we can follow these steps:

### Step 1: Understand the given information
- Focal length (f) of the concave mirror = -50 cm (negative because it is a concave mirror).
- Magnification (m) = -2 (the negative sign indicates that the image is inverted and twice the size of the object).

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

Given that \( m = -2 \), we can write:
\[ -2 = \frac{-v}{u} \]
This simplifies to:
\[ v = 2u \]

### Step 3: Use the mirror formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting \( f = -50 \) cm and \( v = 2u \) into the mirror formula, we get:
\[ \frac{1}{-50} = \frac{1}{2u} + \frac{1}{u} \]

### Step 4: Simplify the equation
To combine the terms on the right side, we find a common denominator:
\[ \frac{1}{2u} + \frac{1}{u} = \frac{1 + 2}{2u} = \frac{3}{2u} \]
Thus, we rewrite the equation:
\[ \frac{1}{-50} = \frac{3}{2u} \]

### Step 5: Cross-multiply to solve for \( u \)
Cross-multiplying gives:
\[ 2u = -150 \]
Therefore:
\[ u = -75 \text{ cm} \]

### Conclusion
The object must be placed at a distance of 75 cm from the mirror (the negative sign indicates that it is on the same side as the object).

### Final Answer
The object distance \( u \) is -75 cm.

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