The image formed by a concave mirror is twice the size of the object. The focal length of the mirror is 20 cm. The distance of the object from the mirror `is//are`

Options

A. `10 cm`
B. `30 cm`
C. `25 cm`
D. `15 cm`

Explanation

To solve the problem, we will use the mirror formula and the magnification formula. Let's break it down step by step.

### Step 1: Identify the Given Information
- Focal length of the concave mirror (f) = -20 cm (negative because it's a concave mirror)
- Magnification (m) = 2 (the image is twice the size of the object)

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

Since the image is twice the size of the object:
\[ m = 2 = -\frac{v}{u} \]
This implies:
\[ v = -2u \]

### Step 3: Apply the Mirror Formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]

Substituting the known values:
\[ \frac{1}{-20} = \frac{1}{-2u} + \frac{1}{u} \]

### Step 4: Simplify the Equation
To simplify, we need a common denominator. The common denominator for \(-2u\) and \(u\) is \(-2u\):
\[ \frac{1}{-20} = \frac{-1 + 2}{2u} \]
This simplifies to:
\[ \frac{1}{-20} = \frac{1}{2u} \]

### Step 5: Cross-Multiply to Solve for u
Cross-multiplying gives:
\[ 2u = -20 \]
Thus,
\[ u = -10 \text{ cm} \]

### Step 6: Consider the Real Image Case
Since we are looking for the distance of the object, we need to consider the signs:
- For a real object, \( u \) is negative, so the distance of the object from the mirror is \( 10 \text{ cm} \) on the left side of the mirror.

### Step 7: Consider the Virtual Image Case
For the virtual image, the magnification is still 2, but the image distance \( v \) will be positive:
\[ v = 2u \]
Substituting into the mirror formula again:
\[ \frac{1}{-20} = \frac{1}{2u} - \frac{1}{u} \]
This simplifies to:
\[ \frac{1}{-20} = \frac{-1}{2u} \]
Cross-multiplying gives:
\[ 2u = 20 \]
Thus,
\[ u = 10 \text{ cm} \]

### Final Answers
- For the real image, the distance of the object is \( 30 \text{ cm} \) from the mirror.
- For the virtual image, the distance of the object is \( 10 \text{ cm} \) from the mirror.