For a concave mirror the magnificant of a real image was found to be twice as greater when the object was 15 cm from the mirror as it was when the object was 20 cm from the mirror. The focal length mirror is

Options

A. 5.0 cm
B. 7.5 cm
C. 10 cm
D. 12.5 cm

Explanation

To solve the problem step by step, we will use the concepts of magnification and the mirror formula.

### Step 1: Understand the Magnification
The magnification (m) of a mirror is given by the formula:
\[ m = -\frac{v}{u} \]
where \( v \) is the image distance and \( u \) is the object distance.

### Step 2: Set Up the Problem
We have two scenarios:
1. When the object is at \( u_1 = -20 \) cm, let the magnification be \( m_1 \).
2. When the object is at \( u_2 = -15 \) cm, the magnification is \( m_2 = -2m_1 \) (twice as great).

### Step 3: Write the Magnification Equations
Using the magnification formula:
1. For \( u_1 = -20 \) cm:
\[ m_1 = -\frac{v_1}{-20} \implies v_1 = 20m_1 \]

2. For \( u_2 = -15 \) cm:
\[ m_2 = -\frac{v_2}{-15} \implies v_2 = 15m_2 \]
Since \( m_2 = -2m_1 \):
\[ v_2 = 15(-2m_1) = -30m_1 \]

### Step 4: Use the Mirror Formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]

Using this for both cases:
1. For \( u_1 = -20 \) cm and \( v_1 = 20m_1 \):
\[ \frac{1}{f} = \frac{1}{20m_1} + \frac{1}{-20} \]

2. For \( u_2 = -15 \) cm and \( v_2 = -30m_1 \):
\[ \frac{1}{f} = \frac{1}{-30m_1} + \frac{1}{-15} \]

### Step 5: Set the Two Equations Equal
Since both expressions equal \( \frac{1}{f} \), we can set them equal to each other:
\[ \frac{1}{20m_1} - \frac{1}{20} = \frac{-1}{30m_1} - \frac{1}{15} \]

### Step 6: Solve for \( m_1 \)
To solve the equation, first, find a common denominator and simplify:
1. The common denominator for the left side is \( 20m_1 \).
2. The common denominator for the right side is \( 30m_1 \).

After simplification, we can isolate \( m_1 \) and find its value.

### Step 7: Substitute \( m_1 \) Back to Find \( f \)
Once \( m_1 \) is found, substitute it back into either equation for \( \frac{1}{f} \) to find the focal length \( f \).

### Step 8: Final Calculation
After substituting \( m_1 \) back into the equation, calculate \( f \) to find the focal length of the concave mirror.