A very small object of length a is kept along the axis of a concave mirror of focal length y at a distance x from its pole. Approximate size of the image is

Options

A. `(a^(2)y^(3))/((y-x)^(2))`
B. `(a^(3)y^(2))/((y-x)^(2))`
C. `(ay^(2))/((y-x)^(2))`
D. `(a^(2)y)/((y-x)^(2))`

Explanation

To find the approximate size of the image formed by a concave mirror when a very small object of length \( a \) is placed at a distance \( x \) from its pole, we can follow these steps:

### Step 1: Use the Mirror Formula
The mirror formula for a concave mirror is given by:
\[ \frac{1}{v} + \frac{1}{u} = \frac{1}{f} \]
where:
- \( v \) is the image distance,
- \( u \) is the object distance (which is negative for concave mirrors),
- \( f \) is the focal length (negative for concave mirrors).

For our case:
- \( u = -x \) (since the object is in front of the mirror),
- \( f = -y \) (since the focal length is negative for concave mirrors).

Substituting these values into the mirror formula gives:
\[ \frac{1}{v} - \frac{1}{x} = -\frac{1}{y} \]

### Step 2: Rearranging the Equation
Rearranging the equation to find \( v \):
\[ \frac{1}{v} = -\frac{1}{y} + \frac{1}{x} \]
\[ \frac{1}{v} = \frac{x - y}{xy} \]
Thus,
\[ v = \frac{xy}{x - y} \]

### Step 3: Differentiate to Find Size of Image
To find the size of the image, we differentiate \( v \) with respect to \( x \):
\[ \frac{dv}{dx} = \frac{d}{dx} \left( \frac{xy}{x - y} \right) \]

Using the quotient rule:
\[ \frac{dv}{dx} = \frac{(x - y)(y) - (xy)(1)}{(x - y)^2} \]
This simplifies to:
\[ \frac{dv}{dx} = \frac{y^2}{(x - y)^2} \]

### Step 4: Relate Size of Image to Size of Object
The size of the image \( \Delta v \) can be related to the size of the object \( \Delta x \) by:
\[ \Delta v = \frac{dv}{dx} \cdot \Delta x \]
Substituting \( \Delta x = a \) (the length of the object):
\[ \Delta v = \frac{y^2}{(x - y)^2} \cdot a \]

### Final Result
Thus, the approximate size of the image is:
\[ \Delta v = \frac{y^2 a}{(x - y)^2} \]

### Conclusion
The approximate size of the image formed by the concave mirror is:
\[ \Delta v = \frac{y^2 a}{(x - y)^2} \]