Question

Let \( u \) and \( v \) be the distances of the object and the image from a lens of focal length \( f \). The correct graphical representation of \( u \) and \( v \) for a convex lens when \( |u| > f \), is:

• A. Linear graph
• B. Inverse graph
• C. Parabolic graph
• D. Hyperbolic graph

Hint:

For a convex lens, the object and image distances follow the lens formula, and the graph between \( u \) and \( v \) will be an inverse curve.

Answer 1

The Correct Option is B

In the context of optics, to find the graphical relationship between the object distance (\(u\)) and the image distance (\(v\)) for a convex lens with focal length (\(f\)), we employ the lens formula, which states:

\(\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\)

This equation rearranges to:

\(uv = f(v+u)\)

or

\(v = \frac{uf}{u-f}\)

Given the requirement \(|u| > f\), it follows that both the object and image distances relate inversely, due to the algebraic form of the equation. The term \((u-f)\) in the denominator ensures that with increasing \(u\), \(v\) does not proportionally increase but rather decreases, highlighting an inverse relationship. This characteristic is depicted graphically as an inverse graph. Therefore, the correct graphical representation of the relationship between \(u\) and \(v\) when \(|u| > f\) for a convex lens is:

Inverse graph

Answer 2

Step 1: Understand the given condition.
We are dealing with a convex lens, for which the object and image distances are denoted by \( u \) and \( v \), respectively, and the focal length of the lens is \( f \). The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u}. \]

Step 2: Rearrange the formula for graph representation.
Rearranging the above equation to express \( v \) in terms of \( u \):
\[ \frac{1}{v} = \frac{1}{f} - \frac{1}{u} \quad \Rightarrow \quad v = \frac{fu}{u - f}. \] This relation shows that \( v \) is not a linear function of \( u \); instead, it is a hyperbolic (inverse) relation.

Step 3: Behavior of the graph for a convex lens.
- When \( |u| > f \), the object lies beyond the focal point on the principal axis.
- As \( u \) increases beyond \( f \), \( v \) decreases, but remains positive on the opposite side of the lens.
- The curve (graph) between \( u \) and \( v \) is an **inverse graph (rectangular hyperbola)** that lies in the first quadrant if we take distances according to the Cartesian sign convention.

Step 4: Conclusion.
The correct graphical representation of \( u \) and \( v \) for a convex lens when \( |u| > f \) is an **inverse graph** showing a hyperbolic relationship between \( u \) and \( v \).

Final Answer:
Inverse graph