The two circles `x^2+y^2=ax and x^2+y^2=c^2(c > 0)` touch each other if (1) `a=2c` (2) `|a|=2c` (3) `2|a|=c` (4) `|a|=c`

Options

A. `|a| = 2c`
B. `2 |a| =c`
C. ` | a| =c`
D. `a = 2c`

Explanation

To determine when the two circles \(x^2 + y^2 = ax\) and \(x^2 + y^2 = c^2\) touch each other, we will analyze their centers and radii.

### Step 1: Rewrite the equations of the circles
The first circle can be rewritten as:
\[ x^2 + y^2 - ax = 0 \]
This can be compared to the standard form of a circle, \((x - h)^2 + (y - k)^2 = r^2\).

### Step 2: Identify the center and radius of the first circle
From the equation \(x^2 + y^2 - ax = 0\), we can identify:
- Center \(C_1\) of the first circle:
\[ \left(\frac{a}{2}, 0\right) \]
- Radius \(r_1\) of the first circle:
\[ r_1 = \frac{a}{2} \]

### Step 3: Identify the center and radius of the second circle
The second circle is given by:
\[ x^2 + y^2 = c^2 \]
This can be rewritten as:
\[ x^2 + y^2 - c^2 = 0 \]
From this, we can identify:
- Center \(C_2\) of the second circle:
\[ (0, 0) \]
- Radius \(r_2\) of the second circle:
\[ r_2 = c \]

### Step 4: Calculate the distance between the centers
The distance \(d\) between the centers \(C_1\) and \(C_2\) is given by:
\[ d = \sqrt{\left(\frac{a}{2} - 0\right)^2 + (0 - 0)^2} = \frac{a}{2} \]

### Step 5: Determine the condition for the circles to touch
For the circles to touch each other, the distance between their centers must equal the sum or the difference of their radii. Since one circle can touch the other internally, we have:
\[ d = r_2 - r_1 \]
Substituting the values we have:
\[ \frac{a}{2} = c - \frac{a}{2} \]

### Step 6: Solve the equation
Rearranging the equation gives:
\[ \frac{a}{2} + \frac{a}{2} = c \]
\[ a = c \]

### Step 7: Consider the absolute value
Since \(c\) is positive, we also need to consider the absolute value of \(a\):
\[ |a| = c \]

### Conclusion
Thus, the condition for the two circles to touch each other is:
\[ |a| = c \]
This corresponds to option (4).