A real inverted image in a concave mirror is represented by (`u`,`v`, `f` are corrdinates)

Options

A.
B.
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D.

Answer

To solve the problem of identifying the correct graph representing the relationship between the coordinates of a real inverted image in a concave mirror, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Mirror Formula**:
The mirror formula for a concave mirror is given by:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]
where \( f \) is the focal length, \( u \) is the object distance (negative for real objects in concave mirrors), and \( v \) is the image distance (negative for real images).

2. **Rearranging the Formula**:
We can rearrange the mirror formula to express it in a different form:
\[ \frac{1}{v} = \frac{1}{f} - \frac{1}{u} \]
Multiplying through by \( f \) gives:
\[ \frac{f}{v} + \frac{f}{u} = 1 \]
This can be rewritten as:
\[ \frac{1}{u} + \frac{1}{v} = \frac{1}{f} \]

3. **Identifying Variables**:
Let's set:
- \( x = \frac{1}{u} \)
- \( y = \frac{1}{v} \)
Thus, the equation becomes:
\[ x + y = \frac{1}{f} \]

4. **Graph Interpretation**:
The equation \( x + y = \frac{1}{f} \) represents a straight line in the \( xy \)-plane. However, since we are dealing with a hyperbolic relationship due to the nature of the mirror, we need to consider the behavior of \( x \) and \( y \) as they approach their limits.

5. **Behavior of the Graph**:
- If \( x \) (which corresponds to \( \frac{1}{u} \)) approaches 0 (i.e., \( u \) approaches infinity), then \( y \) (which corresponds to \( \frac{1}{v} \)) approaches \( \frac{1}{f} \).
- Conversely, if \( y \) approaches 0 (i.e., \( v \) approaches infinity), then \( x \) approaches \( \frac{1}{f} \).

6. **Conclusion on Graph**:
The resulting graph will be a hyperbola that opens towards the positive axes, indicating the relationship between \( u \) and \( v \) for real inverted images in a concave mirror.

7. **Selecting the Correct Graph**:
After analyzing the options provided, the correct graph that represents this relationship is the one that depicts a hyperbola in the first quadrant, confirming that both \( u \) and \( v \) are negative for real images.