Consider the following statements :
I . The distance between the lines y = mx `+ c_(1)` and `y = mx + c_(2)` is `(|c_(1) - c_(2)|)/(sqrt(1 -m^(2)))`
II . The distance between the lines `ax + by + c_(1) = 0` and `ax + by + c_(2) = 0` is `(|c_(1) - c_2|)/(sqrt(a^(2) + b^(2)))`
III . The distance between the lines x = c and x = `c_(2)` is `|c_(1) - c_(2)|`
Which of the above statements are correct ?

Options

A. I and II only
B. II and III only
C. I and III only
D. I , II and III only

Answer

To determine which of the statements regarding the distance between lines are correct, we will analyze each statement step by step.

### Statement I:
**The distance between the lines \( y = mx + c_1 \) and \( y = mx + c_2 \) is given by:**
\[ \frac{|c_1 - c_2|}{\sqrt{1 - m^2}} \]

**Step 1:** Identify the lines.
- The two lines are \( y = mx + c_1 \) and \( y = mx + c_2 \).

**Step 2:** Understand the distance formula for parallel lines.
- The general formula for the distance \( d \) between two parallel lines \( y = mx + c_1 \) and \( y = mx + c_2 \) is:
\[ d = \frac{|c_1 - c_2|}{\sqrt{1 + m^2}} \]

**Step 3:** Compare the given formula with the standard formula.
- The formula provided in the statement has \( \sqrt{1 - m^2} \) instead of \( \sqrt{1 + m^2} \).
- Therefore, **Statement I is incorrect.**

### Statement II:
**The distance between the lines \( ax + by + c_1 = 0 \) and \( ax + by + c_2 = 0 \) is given by:**
\[ \frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}} \]

**Step 1:** Identify the lines.
- The two lines are \( ax + by + c_1 = 0 \) and \( ax + by + c_2 = 0 \).

**Step 2:** Understand the distance formula for parallel lines.
- The formula for the distance \( d \) between two parallel lines of the form \( ax + by + c_1 = 0 \) and \( ax + by + c_2 = 0 \) is:
\[ d = \frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}} \]

**Step 3:** Compare the given formula with the standard formula.
- The formula provided in the statement matches the standard formula.
- Therefore, **Statement II is correct.**

### Statement III:
**The distance between the lines \( x = c_1 \) and \( x = c_2 \) is given by:**
\[ |c_1 - c_2| \]

**Step 1:** Identify the lines.
- The two lines are vertical lines \( x = c_1 \) and \( x = c_2 \).

**Step 2:** Understand the distance between vertical lines.
- The distance \( d \) between two vertical lines \( x = c_1 \) and \( x = c_2 \) is simply the absolute difference of their constants:
\[ d = |c_1 - c_2| \]

**Step 3:** Compare the given formula with the standard formula.
- The formula provided in the statement matches the standard formula.
- Therefore, **Statement III is correct.**

### Conclusion:
- **Statement I is incorrect.**
- **Statements II and III are correct.**

### Final Answer:
The correct statements are II and III.

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