Object is placed 15 cm from a concave mirror of focal length 10 cm, then the nature of the image formed will be

Options

A. magnified and inverted
B. magnified and erect
C. small in size and inverted
D. small in size and erect

Explanation

To determine the nature of the image formed by a concave mirror when an object is placed 15 cm from it, we can use the mirror formula and the concept of magnification. Here’s a step-by-step solution:

### Step 1: Identify the given values
- Object distance (u) = -15 cm (the object distance is taken as negative in mirror conventions)
- Focal length (f) = -10 cm (the focal length of a concave mirror is negative)

### Step 2: Use the mirror formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Rearranging this formula to find the image distance (v):
\[ \frac{1}{v} = \frac{1}{f} - \frac{1}{u} \]

### Step 3: Substitute the values into the formula
Substituting the known values into the rearranged mirror formula:
\[ \frac{1}{v} = \frac{1}{-10} - \frac{1}{-15} \]
Finding a common denominator (which is 30):
\[ \frac{1}{v} = -\frac{3}{30} + \frac{2}{30} = -\frac{1}{30} \]

### Step 4: Calculate v
Taking the reciprocal to find v:
\[ v = -30 \text{ cm} \]
The negative sign indicates that the image is formed on the same side as the object, which is characteristic of real images in concave mirrors.

### Step 5: Determine the nature of the image
To find the nature of the image, we analyze the sign of v:
- Since v is negative, the image is real.
- We also need to check the magnification (m) to determine if the image is inverted or upright.

### Step 6: Calculate the magnification
The magnification (m) for mirrors is given by:
\[ m = -\frac{v}{u} \]
Substituting the values:
\[ m = -\frac{-30}{-15} = -2 \]
The negative sign indicates that the image is inverted, and the magnitude (2) indicates that the image is magnified (since |m| > 1).

### Conclusion
The nature of the image formed is:
- Real
- Inverted
- Magnified

### Final Answer
The image is real, inverted, and magnified.

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