Statement 1 `:` The number of common tangents to the circles `x^(2) +y^(2) -x =0` and `x^(2) +y^(2) +x =0 ` is 3.
Statement 2 `:` If two circles touch each other externally then it has two direct common tangents and one indirect common tangent.

Options

A. Statement-1 `:` is True, Statement-2 is True and Statement-2 is a correct explanation for Statement-1
B. Statement-1 is True, Statement-2 is True and Statement-2 is NOT a correct explanation for Statement -1
C. Statement-1 is True, Statement -2 is False
D. Statement-1 is False, Statement-2 is True

Answer

To solve the problem, we need to analyze the two statements regarding the number of common tangents to the given circles.

### Step 1: Identify the equations of the circles
The equations of the circles given are:
1. \( x^2 + y^2 - x = 0 \)
2. \( x^2 + y^2 + x = 0 \)

### Step 2: Rewrite the equations in standard form
To rewrite these equations in standard form, we can complete the square.

For the first circle:
\[ x^2 - x + y^2 = 0 \implies (x - \frac{1}{2})^2 + y^2 = \frac{1}{4} \]
This represents a circle with center \( C_1 = \left(\frac{1}{2}, 0\right) \) and radius \( r_1 = \frac{1}{2} \).

For the second circle:
\[ x^2 + x + y^2 = 0 \implies (x + \frac{1}{2})^2 + y^2 = \frac{1}{4} \]
This represents a circle with center \( C_2 = \left(-\frac{1}{2}, 0\right) \) and radius \( r_2 = \frac{1}{2} \).

### Step 3: Find the distance between the centers of the circles
The distance \( d \) between the centers \( C_1 \) and \( C_2 \) is calculated as follows:
\[ d = \sqrt{\left(\frac{1}{2} - \left(-\frac{1}{2}\right)\right)^2 + (0 - 0)^2} = \sqrt{(1)^2} = 1 \]

### Step 4: Check the condition for external tangency
The circles touch each other externally if the distance between their centers \( d \) is equal to the sum of their radii \( r_1 + r_2 \):
\[ r_1 + r_2 = \frac{1}{2} + \frac{1}{2} = 1 \]
Since \( d = 1 \) and \( r_1 + r_2 = 1 \), the circles touch each other externally.

### Step 5: Determine the number of common tangents
According to the properties of tangents:
- If two circles touch externally, they have 3 common tangents: 2 direct and 1 indirect.

Thus, the first statement is true: the number of common tangents to the circles is 3.

### Step 6: Verify the second statement
The second statement claims that if two circles touch each other externally, then they have 2 direct common tangents and 1 indirect common tangent. This is also true based on the properties of tangents.

### Conclusion
Both statements are true, and statement 2 correctly explains statement 1.