The two circles `x^(2)+y^(2)-cx=0 and x^(2)+y^(2)=4` touch each other if:

Options

A. `|c|=4`
B. `|c|=2`
C. `|c|=8`
D. None of these

Explanation

To determine the condition under which the two circles \(x^2 + y^2 - cx = 0\) and \(x^2 + y^2 = 4\) touch each other, we will follow these steps:

### Step 1: Identify the centers and radii of the circles

1. **Circle 1**: The equation \(x^2 + y^2 - cx = 0\) can be rewritten as:
\[ x^2 + y^2 = cx \]
This represents a circle with center \(\left(\frac{c}{2}, 0\right)\) and radius \(r_1 = \frac{c}{2}\).

2. **Circle 2**: The equation \(x^2 + y^2 = 4\) represents a circle with center \((0, 0)\) and radius \(r_2 = 2\).

### Step 2: Calculate the distance between the centers

The distance \(d\) between the centers of the two circles is given by:
\[ d = \sqrt{\left(0 - \frac{c}{2}\right)^2 + (0 - 0)^2} = \left|\frac{c}{2}\right| \]

### Step 3: Set up the conditions for the circles to touch

The two circles can touch each other in two ways:
1. Externally: The distance between the centers equals the sum of the radii:
\[ d = r_1 + r_2 \]
Substituting the values:
\[ \left|\frac{c}{2}\right| = \frac{c}{2} + 2 \]

2. Internally: The distance between the centers equals the absolute difference of the radii:
\[ d = |r_1 - r_2| \]
Substituting the values:
\[ \left|\frac{c}{2}\right| = \left|\frac{c}{2} - 2\right| \]

### Step 4: Solve the external tangency condition

From the external tangency condition:
\[ \left|\frac{c}{2}\right| = \frac{c}{2} + 2 \]

**Case 1**: \( \frac{c}{2} \geq 0 \)
\[ \frac{c}{2} = \frac{c}{2} + 2 \implies 0 = 2 \quad \text{(no solution)} \]

**Case 2**: \( \frac{c}{2} < 0 \)
\[ -\frac{c}{2} = \frac{c}{2} + 2 \implies -c = c + 4 \implies -2c = 4 \implies c = -2 \]

### Step 5: Solve the internal tangency condition

From the internal tangency condition:
\[ \left|\frac{c}{2}\right| = \left|\frac{c}{2} - 2\right| \]

**Case 1**: \( \frac{c}{2} \geq 2 \)
\[ \frac{c}{2} = \frac{c}{2} - 2 \implies 0 = -2 \quad \text{(no solution)} \]

**Case 2**: \( \frac{c}{2} < 2 \)
\[ -\frac{c}{2} = \frac{c}{2} - 2 \implies -c = c - 4 \implies -2c = -4 \implies c = 2 \]

### Step 6: Combine the results

The values of \(c\) for which the circles touch each other are:
- From external tangency: \(c = -2\)
- From internal tangency: \(c = 2\)

Thus, the final condition is:
\[ |c| = 2 \]

### Final Answer

The two circles touch each other if:
\[ \boxed{|c| = 2} \]