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. 60 cm
C. 125 cm
D. 50 cm

Explanation

To solve the problem, we need to find the object distance (u) for a concave mirror given its focal length (f) and the magnification (m). Here are the steps to arrive at the solution:

### Step 1: Identify the given values
- Focal length of the concave mirror, \( f = -50 \) cm (negative because it is a concave mirror).
- Magnification, \( m = -2 \) (since the image is inverted and magnified).

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

Since we know \( m = -2 \), we can write:
\[ -2 = -\frac{v}{u} \]
This simplifies to:
\[ 2 = \frac{v}{u} \quad \text{or} \quad v = 2u \]

### Step 3: Use the mirror formula
The mirror formula relates the focal length (f), object distance (u), and image distance (v):
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting \( f = -50 \) cm and \( v = 2u \) into the mirror formula gives:
\[ \frac{1}{-50} = \frac{1}{2u} + \frac{1}{u} \]

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

### Step 5: Cross-multiply to solve for u
Cross-multiplying gives:
\[ 1 \cdot 2u = -50 \cdot 3 \]
This simplifies to:
\[ 2u = -150 \]
Dividing both sides by 2:
\[ u = -75 \text{ cm} \]

### Step 6: Conclusion
The object should be placed at a distance of 75 cm in front of the concave mirror (the negative sign indicates the direction).

### Final Answer
The object distance \( u = -75 \) cm, which means the object should be placed 75 cm in front of the mirror.

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