A graph is drawn with `(1)/(u)` along x-axis and `(1)/(v)` along the y-axis. If the intercept on the x-axis is `0.5 m^-1`, the focal length of the mirror is (in meter).

Options

A. (a) 2 .00
B. (b)0. 50
C. (c)0 .20
D. (d)1 .00

Explanation

To solve the problem, we need to find the focal length of the mirror using the information provided about the graph of \( \frac{1}{u} \) and \( \frac{1}{v} \).

### Step-by-Step Solution:

1. **Understand the Mirror Equation**:
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Rearranging this, we can express \( \frac{1}{v} \) in terms of \( \frac{1}{u} \):
\[ \frac{1}{v} = -\frac{1}{u} + \frac{1}{f} \]

2. **Identify the Graph Parameters**:
In the graph, \( \frac{1}{u} \) is plotted on the x-axis and \( \frac{1}{v} \) on the y-axis. The equation can be compared to the linear equation \( y = mx + c \), where:
- \( y = \frac{1}{v} \)
- \( x = \frac{1}{u} \)
- Slope \( m = -1 \)
- Intercept \( c = \frac{1}{f} \)

3. **Use the X-Intercept**:
The x-intercept of the graph is given as \( 0.5 \, \text{m}^{-1} \). At the x-intercept, \( \frac{1}{v} = 0 \), which means:
\[ \frac{1}{u} = 0.5 \, \text{m}^{-1} \]

4. **Calculate \( u \)**:
From the x-intercept:
\[ u = \frac{1}{0.5} = 2 \, \text{m} \]

5. **Substitute \( u \) into the Mirror Equation**:
Now substitute \( u \) back into the rearranged mirror equation:
\[ 0 = -\frac{1}{2} + \frac{1}{f} \]
Rearranging gives:
\[ \frac{1}{f} = \frac{1}{2} \]

6. **Calculate the Focal Length \( f \)**:
Taking the reciprocal:
\[ f = 2 \, \text{m} \]

### Final Answer:
The focal length of the mirror is \( 2 \, \text{m} \).

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