The number of common tangents to the circles `x^(2)+y^(2) -x=0, x^(2)+y^(2)+x=0` is

Options

A. 2
B. 1
C. 4
D. 3

Answer

To determine the number of common tangents to the circles given by the equations \(x^2 + y^2 - x = 0\) and \(x^2 + y^2 + x = 0\), we will follow these steps:

### Step 1: Rewrite the equations of the circles
The equations of the circles can be rewritten in standard form.

1. For the first circle:
\[ x^2 + y^2 - x = 0 \implies x^2 - x + y^2 = 0 \]
Completing the square for \(x\):
\[ (x - \frac{1}{2})^2 + y^2 = \frac{1}{4} \]
This gives us:
- Center \(C_1 = \left(\frac{1}{2}, 0\right)\)
- Radius \(r_1 = \frac{1}{2}\)

2. For the second circle:
\[ x^2 + y^2 + x = 0 \implies x^2 + x + y^2 = 0 \]
Completing the square for \(x\):
\[ (x + \frac{1}{2})^2 + y^2 = \frac{1}{4} \]
This gives us:
- Center \(C_2 = \left(-\frac{1}{2}, 0\right)\)
- Radius \(r_2 = \frac{1}{2}\)

### Step 2: Calculate the distance between the centers
Now, we calculate the distance \(d\) between the centers \(C_1\) and \(C_2\):
\[ d = \sqrt{ \left(\frac{1}{2} - \left(-\frac{1}{2}\right)\right)^2 + (0 - 0)^2 } = \sqrt{ \left(\frac{1}{2} + \frac{1}{2}\right)^2 } = \sqrt{1^2} = 1 \]

### Step 3: Calculate the sum of the radii
Next, we calculate the sum of the radii:
\[ r_1 + r_2 = \frac{1}{2} + \frac{1}{2} = 1 \]

### Step 4: Determine the relationship between the distance and the radii
Now we compare the distance \(d\) with the sum of the radii \(r_1 + r_2\):
- Since \(d = r_1 + r_2 = 1\), it indicates that the circles touch externally.

### Step 5: Determine the number of common tangents
When two circles touch externally, the number of common tangents is 3 (2 external tangents and 1 internal tangent).

### Conclusion
Thus, the number of common tangents to the circles is:
\[ \text{Number of common tangents} = 3 \]