An concave mirror of focal length 30 cm forms an image of double of the height of object. If the image is real then the distance of the object from image is

Options

A. 45 cm
B. 30 cm
C. 20 cm
D. 90 cm

Explanation

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

### Step 1: Identify the given data
- Focal length (f) of the concave mirror = -30 cm (the focal length is negative for concave mirrors)
- The image formed is double the height of the object, which means the magnification (m) = -2 (since the image is real and inverted).

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

### Step 3: Substitute the known values into the magnification formula
Since \( m = -2 \):
\[ -2 = -\frac{v}{u} \]
This simplifies to:
\[ v = 2u \]

### Step 4: Use the mirror formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting \( f = -30 \) cm and \( v = 2u \):
\[ \frac{1}{-30} = \frac{1}{2u} + \frac{1}{u} \]

### Step 5: Simplify the equation
Combine the terms on the right:
\[ \frac{1}{-30} = \frac{1 + 2}{2u} \]
\[ \frac{1}{-30} = \frac{3}{2u} \]

### Step 6: Cross-multiply to solve for u
Cross-multiplying gives:
\[ 3(-30) = 2u \]
\[ -90 = 2u \]
\[ u = -45 \text{ cm} \]

### Step 7: Find the distance of the object from the image
The distance between the object and the image (d) can be calculated as:
\[ d = |u| + |v| \]
Since \( v = 2u \):
\[ v = 2(-45) = -90 \text{ cm} \]
Thus:
\[ d = |-45| + |-90| = 45 + 90 = 135 \text{ cm} \]

### Final Answer
The distance of the object from the image is **135 cm**.

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