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The question involves understanding the relationship between the object distance $u$, the image distance $v$, and the focal length $f$ of a convex lens. The lens equation is given by:
$$ \frac{1}{v} - \frac{1}{u} = \frac{1}{f} $$
### Step 1: Understanding the Lens Equation
The lens equation relates the object distance ($u$), the image distance ($v$), and the focal length ($f$) of the lens. For a convex lens, the focal length is positive. This equation indicates that as the object distance $u$ changes, the image distance $v$ will also change in a specific manner.
### Step 2: Rearranging the Lens Equation
We can rearrange the lens equation to express $v$ in terms of $u$:
$$ \frac{1}{v} = \frac{1}{u} + \frac{1}{f} $$
Taking the reciprocal gives:
$$ v = \frac{uf}{u + f} $$
### Step 3: Analyzing the Graph
Now, we need to analyze how this relationship translates into a graph. The equation $v = \frac{uf}{u + f}$ suggests a hyperbolic relationship between $u$ and $v$.
- As $u$ increases, $v$ will also increase, but at a decreasing rate. This means that the graph will start steep and then flatten out as $u$ becomes very large.
- When $u$ approaches $f$, $v$ will approach infinity, indicating that the image moves further away as the object comes closer to the focal point.
### Step 4: Conclusion
The graph that best represents this relationship is a hyperbola, which is consistent with the behavior of the lens equation. The correct graph will show that as $u$ increases, $v$ also increases, but not linearly.
Thus, the correct answer is the graph that depicts this hyperbolic relationship between $u$ and $v$ for a convex lens.
