Find the distance of object form a concave morror of focal length 10 cm so that image size is four times the size of the object.

Explanation

To find the distance of the object from a concave mirror of focal length 10 cm such that the image size is four times the size of the object, we can follow these steps:

### Step 1: Understand the relationship between object distance (u), image distance (v), and magnification (m).
The magnification (m) for mirrors is given by the formula:
\[ m = \frac{h'}{h} = -\frac{v}{u} \]
where:
- \( h' \) is the height of the image,
- \( h \) is the height of the object,
- \( v \) is the image distance,
- \( u \) is the object distance.

Given that the image size is four times the object size, we can write:
\[ m = 4 \]
Thus, we have:
\[ -\frac{v}{u} = 4 \]
This implies:
\[ v = -4u \]

### Step 2: Use the mirror formula.
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Here, the focal length \( f \) of the concave mirror is given as -10 cm (negative because it is a concave mirror).

### Step 3: Substitute \( v \) in the mirror formula.
Substituting \( v = -4u \) into the mirror formula gives:
\[ \frac{1}{-10} = \frac{1}{-4u} + \frac{1}{u} \]

### Step 4: Simplify the equation.
To simplify, find a common denominator:
\[ \frac{1}{-10} = \frac{-1 + 4}{4u} \]
\[ \frac{1}{-10} = \frac{3}{4u} \]

### Step 5: Cross-multiply to solve for \( u \).
Cross-multiplying gives:
\[ 3 \cdot (-10) = 4u \]
\[ -30 = 4u \]
Thus:
\[ u = -\frac{30}{4} = -7.5 \text{ cm} \]

### Step 6: Calculate the distance for the second case (virtual image).
For a virtual image, the magnification is still 4, but the image distance \( v \) will be positive:
\[ v = 4u \]
Substituting this into the mirror formula:
\[ \frac{1}{-10} = \frac{1}{4u} + \frac{1}{u} \]
This leads to:
\[ \frac{1}{-10} = \frac{5}{4u} \]
Cross-multiplying gives:
\[ 5 \cdot (-10) = 4u \]
\[ -50 = 4u \]
Thus:
\[ u = -\frac{50}{4} = -12.5 \text{ cm} \]

### Final Results:
1. For a real image, the object distance \( u \) is \( -7.5 \) cm.
2. For a virtual image, the object distance \( u \) is \( -12.5 \) cm.