If the distance between the virtual image from Its real object is 60 cm in case of a concave mirror . The focal length of the concave mirror If the image formed is 3 times magnified image is

Answer

To solve the problem, we need to find the focal length of a concave mirror given the distance between the virtual image and the real object, and the magnification of the image. Here are the step-by-step calculations:

### Step 1: Understand the given information
We know:
- The distance between the virtual image and the real object is 60 cm.
- The magnification (m) of the image is 3 (which means the image is three times the size of the object).

### Step 2: Define the variables
Let:
- \( u \) = object distance (distance from the mirror to the object)
- \( v \) = image distance (distance from the mirror to the image)

Since the image is virtual, it is formed behind the mirror, so:
- \( v \) will be positive (as per the sign convention).
- \( u \) will be negative (as the object is in front of the mirror).

From the problem, we have:
\[ |u| + v = 60 \text{ cm} \]

### Step 3: Use the magnification formula
The magnification (m) is given by:
\[ m = -\frac{v}{u} \]

Since the magnification is positive (the image is erect), we have:
\[ 3 = -\frac{v}{u} \]
This implies:
\[ v = -3u \]

### Step 4: Substitute \( v \) in the distance equation
Now we can substitute \( v \) in the distance equation:
\[ |u| + (-3u) = 60 \]
Since \( u \) is negative, we can write:
\[ -u - 3u = 60 \]
\[ -4u = 60 \]
\[ u = -15 \text{ cm} \]

### Step 5: Find \( v \)
Now, substitute \( u \) back to find \( v \):
\[ v = -3(-15) = 45 \text{ cm} \]

### Step 6: Use the mirror formula to find the focal length
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]

Substituting the values of \( v \) and \( u \):
\[ \frac{1}{f} = \frac{1}{45} + \frac{1}{-15} \]

### Step 7: Calculate \( \frac{1}{f} \)
Finding a common denominator:
\[ \frac{1}{f} = \frac{1}{45} - \frac{3}{45} \]
\[ \frac{1}{f} = \frac{-2}{45} \]

### Step 8: Find \( f \)
Taking the reciprocal to find \( f \):
\[ f = -\frac{45}{2} = -22.5 \text{ cm} \]

### Conclusion
The focal length of the concave mirror is \( -22.5 \) cm. The negative sign indicates that the focal point is on the same side as the object, which is consistent with the properties of concave mirrors.