The two circles `x^(2)+y^(2)=ax and x^(2)+y^(2)=c^(2)(c gt 0)` touch each other, if `|(c )/(a )|` is equal to

Options

A. 2
B. `(1)/(2)`
C. 1
D. None of these

Answer

To solve the problem of determining the value of \(|\frac{c}{a}|\) when the two circles \(x^2 + y^2 = ax\) and \(x^2 + y^2 = c^2\) touch each other, we will follow these steps:

### Step 1: Identify the equations of the circles
The first circle is given by the equation:
\[ x^2 + y^2 = ax \]
We can rewrite this as:
\[ x^2 - ax + y^2 = 0 \]

The second circle is given by the equation:
\[ x^2 + y^2 = c^2 \]

### Step 2: Rewrite the first circle in standard form
To convert the first circle into standard form, we complete the square for the \(x\) terms:
\[ x^2 - ax + \left(\frac{a}{2}\right)^2 - \left(\frac{a}{2}\right)^2 + y^2 = 0 \]
This simplifies to:
\[ \left(x - \frac{a}{2}\right)^2 + y^2 = \left(\frac{a}{2}\right)^2 \]
From this, we can identify:
- Center of the first circle: \(\left(\frac{a}{2}, 0\right)\)
- Radius of the first circle: \(\frac{a}{2}\)

### Step 3: Identify the second circle's parameters
For the second circle:
- Center: \((0, 0)\)
- Radius: \(|c|\) (since \(c > 0\), we can simply use \(c\))

### Step 4: Set up the condition for the circles touching
The circles touch each other if the distance between their centers equals the sum or difference of their radii. The distance \(d\) between the centers is:
\[ d = \sqrt{\left(0 - \frac{a}{2}\right)^2 + (0 - 0)^2} = \frac{a}{2} \]

The condition for the circles to touch externally is:
\[ d = r_1 + r_2 \]
Thus:
\[ \frac{a}{2} = \frac{a}{2} + c \]
This simplifies to:
\[ 0 = c \quad \text{(not possible since \(c > 0\))} \]

For internal touching, we have:
\[ d = |r_1 - r_2| \]
Thus:
\[ \frac{a}{2} = \left|\frac{a}{2} - c\right| \]

### Step 5: Solve the equation for internal touching
This leads to two cases:
1. \(\frac{a}{2} - c = \frac{a}{2}\) (not possible)
2. \(\frac{a}{2} - c = -\frac{a}{2}\)

From the second case:
\[ \frac{a}{2} - c = -\frac{a}{2} \]
This simplifies to:
\[ c = a \]

### Step 6: Find \(|\frac{c}{a}|\)
Now substituting \(c = a\):
\[ |\frac{c}{a}| = |\frac{a}{a}| = 1 \]

### Conclusion
Thus, the value of \(|\frac{c}{a}|\) when the circles touch each other is:
\[ \boxed{1} \]