Distance between a candle and screen is D. A convex lens of focal length 20 cm is brought in between them and it is found that real image of candle can be formed on the screen for some position of lens.

Options

A. D must be less than or equal to 80 cm
B. D must be greater than or equal to 80 cm
C. D may be less than 160 cm
D. D may be greater than 160 cm

Explanation

To solve the problem, we need to analyze the situation involving a candle, a convex lens, and a screen. The distance between the candle and the screen is given as \( D \), and the focal length of the convex lens is \( f = 20 \, \text{cm} \). We want to determine the possible values of \( D \) for which a real image of the candle can be formed on the screen.

### Step-by-Step Solution:

1. **Understanding the Lens Formula**:
The lens formula relates the object distance \( u \), image distance \( v \), and focal length \( f \) of a lens:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]

2. **Identifying Object and Image Positions**:
- Let the distance from the candle to the lens be \( u \).
- The distance from the lens to the screen will then be \( D - u \).
- The image distance \( v \) can be expressed as \( D - u \).

3. **Substituting into the Lens Formula**:
Substitute \( v \) into the lens formula:
\[ \frac{1}{20} = \frac{1}{D - u} - \frac{1}{u} \]

4. **Rearranging the Equation**:
Rearranging gives:
\[ \frac{1}{D - u} = \frac{1}{20} + \frac{1}{u} \]
This can be further simplified to:
\[ \frac{1}{D - u} = \frac{u + 20}{20u} \]

5. **Cross-Multiplying**:
Cross-multiplying yields:
\[ 20u = (D - u)(u + 20) \]

6. **Expanding and Rearranging**:
Expanding the right side:
\[ 20u = Du + 20D - u^2 - 20u \]
Rearranging gives:
\[ u^2 + (D - 40)u + 20D = 0 \]

7. **Using the Discriminant**:
For the quadratic equation \( u^2 + (D - 40)u + 20D = 0 \) to have real solutions, the discriminant must be non-negative:
\[ (D - 40)^2 - 80D \geq 0 \]

8. **Simplifying the Discriminant**:
Simplifying the discriminant:
\[ D^2 - 80D + 1600 \geq 0 \]
This can be factored or solved using the quadratic formula.

9. **Finding the Roots**:
The roots of the equation \( D^2 - 80D + 1600 = 0 \) are:
\[ D = \frac{80 \pm \sqrt{0}}{2} = 40 \]
Since the discriminant is zero, the equation has a double root at \( D = 80 \).

10. **Analyzing the Inequality**:
The quadratic opens upwards, so the solution to the inequality is:
\[ D \leq 80 \quad \text{or} \quad D \geq 80 \]
Thus, \( D \) can be any value greater than or equal to \( 80 \, \text{cm} \).

### Conclusion:
The distance \( D \) between the candle and the screen must be greater than or equal to \( 80 \, \text{cm} \) for a real image to be formed on the screen by the convex lens.