A plot of modulus of image distance versus object distance for a spherical mirror is a:

Options

A. Straight Line
B. Circle
C. Parabola
D. Hyperbola

Explanation

To solve the question regarding the plot of the modulus of image distance versus object distance for a spherical mirror, we will follow these steps:

### Step 1: Understand the Mirror Formula
The mirror formula relates the object distance (u), image distance (v), and focal length (f) of a spherical mirror:
\[ \frac{1}{v} + \frac{1}{u} = \frac{1}{f} \]
From this, we can express \( v \) in terms of \( u \):
\[ \frac{1}{v} = \frac{1}{f} - \frac{1}{u} \]
This can be rearranged to:
\[ v = \frac{fu}{u - f} \]

### Step 2: Analyze the Equation
The equation \( v = \frac{fu}{u - f} \) indicates that as \( u \) approaches \( f \), \( v \) approaches infinity. This suggests a hyperbolic relationship because the image distance \( v \) will not be defined when \( u = f \).

### Step 3: Consider the Modulus
Since the question asks for the modulus of the image distance, we will consider \( |v| \). The relationship remains:
\[ |v| = \left| \frac{fu}{u - f} \right| \]
This means that the graph will still exhibit hyperbolic characteristics, as the absolute value does not change the nature of the relationship.

### Step 4: Plot the Graph
To plot the graph of \( |v| \) versus \( u \):
- When \( u = 0 \), \( |v| \) is undefined (as \( v \) approaches infinity).
- When \( u = f \), \( |v| \) is also undefined (as \( v \) approaches infinity).
- For values of \( u \) less than \( f \), \( |v| \) will be negative, but since we are taking the modulus, it will be positive.
- For values of \( u \) greater than \( f \), \( |v| \) will be positive.

### Step 5: Conclusion
The plot of \( |v| \) versus \( u \) will resemble a hyperbola, with branches in the first and second quadrants, indicating that as the object distance increases or decreases away from the focal point, the image distance behaves hyperbolically.

### Final Answer
The plot of the modulus of image distance versus object distance for a spherical mirror is a hyperbola.

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