A concave mirror forms a real image four times as tall as the object placed 10 cm infront of mirror. Find the position of the image and the radius of curvature of the mirror.

Answer

To solve the problem step by step, we will use the mirror formula and the magnification formula for a concave mirror.

### Step 1: Understand the given information
- The object distance (u) is given as 10 cm in front of the mirror. According to the sign convention for mirrors, this will be taken as negative:
\[ u = -10 \, \text{cm} \]
- The magnification (m) is given as 4 times the height of the object. Since the image is real and inverted, the magnification will be negative:
\[ m = -4 \]

### Step 2: Use the magnification formula
The magnification (m) for mirrors is given by the formula:
\[ m = -\frac{V}{U} \]
Where:
- \( V \) is the image distance
- \( U \) is the object distance

Substituting the known values:
\[ -4 = -\frac{V}{-10} \]

### Step 3: Solve for the image distance (V)
Rearranging the equation to solve for \( V \):
\[ -4 = \frac{V}{10} \]
Multiplying both sides by -10:
\[ V = -40 \, \text{cm} \]

### Step 4: Interpret the image distance
The negative sign indicates that the image is formed on the same side as the object, which is typical for a real image formed by a concave mirror.

### Step 5: Use the mirror formula to find the focal length (F)
The mirror formula is given by:
\[ \frac{1}{F} = \frac{1}{V} + \frac{1}{U} \]
Substituting the values of \( V \) and \( U \):
\[ \frac{1}{F} = \frac{1}{-40} + \frac{1}{-10} \]

### Step 6: Calculate the focal length (F)
Finding a common denominator (which is -40):
\[ \frac{1}{F} = \frac{1}{-40} + \frac{4}{-40} = \frac{-5}{40} \]
Thus,
\[ F = \frac{-40}{5} = -8 \, \text{cm} \]

### Step 7: Find the radius of curvature (R)
The radius of curvature (R) is related to the focal length (F) by the formula:
\[ R = 2F \]
Substituting the value of \( F \):
\[ R = 2 \times -8 = -16 \, \text{cm} \]

### Final Answers
- The position of the image (V) is \( -40 \, \text{cm} \).
- The radius of curvature (R) is \( -16 \, \text{cm} \).

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