The distance between an object and its doubly magnified image by a concave mirror is: [Assume `f`= focal length]

Options

A. `3 f//2`
B. `2 f//3`
C. `3 f`
D. depends on whether the image is real or virtual.

Explanation

To solve the problem of finding the distance between an object and its doubly magnified image by a concave mirror, we can follow these steps:

### Step 1: Understand the Magnification
The problem states that the image is doubly magnified. This means that the magnification (M) is given by:
\[ M = 2 \]

### Step 2: Use the Magnification Formula
For mirrors, the magnification is given by the formula:
\[ M = -\frac{v}{u} \]
where \( v \) is the image distance and \( u \) is the object distance. Since the magnification is 2, we can write:
\[ 2 = -\frac{v}{u} \]
From this, we can express \( v \) in terms of \( u \):
\[ v = -2u \]

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

### Step 4: Simplify the Equation
To combine the fractions on the right side, we need a common denominator:
\[ \frac{1}{f} = \frac{-1 + 2}{2u} = \frac{1}{2u} \]
Thus, we have:
\[ \frac{1}{f} = \frac{1}{2u} \]

### Step 5: Solve for \( u \)
Now, we can solve for \( u \):
\[ 2u = f \]
\[ u = \frac{f}{2} \]

### Step 6: Find the Distance Between Object and Image
The distance \( d \) between the object and the image is given by:
\[ d = |u| + |v| \]
Substituting \( v = -2u \):
\[ d = |u| + |-2u| = |u| + 2|u| = 3|u| \]
Now substituting \( u = \frac{f}{2} \):
\[ d = 3 \left(\frac{f}{2}\right) = \frac{3f}{2} \]

### Final Answer
Thus, the distance between the object and its doubly magnified image by a concave mirror is:
\[ \frac{3f}{2} \]