The two circles `x^(2)+y^(2)=ax, x^(2)+y^(2)=c^(2) (c gt 0)` touch each other if

Options

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

Answer

To determine the condition under which the two circles \( x^2 + y^2 = ax \) and \( x^2 + y^2 = c^2 \) (where \( c > 0 \)) touch each other, we can follow these steps:

### Step 1: Rewrite the equations of the circles
The first circle can be rewritten in standard form:
\[ x^2 + y^2 = ax \implies x^2 - ax + y^2 = 0 \]
This can be rearranged to:
\[ x^2 - ax + y^2 = 0 \]

### Step 2: Complete the square for the first circle
To complete the square for the \( x \) terms, we have:
\[ x^2 - ax = \left(x - \frac{a}{2}\right)^2 - \left(\frac{a}{2}\right)^2 \]
Thus, the equation becomes:
\[ \left(x - \frac{a}{2}\right)^2 + y^2 = \left(\frac{a}{2}\right)^2 \]
From this, we can identify the center and radius of the first circle:
- Center \( C_1 = \left(\frac{a}{2}, 0\right) \)
- Radius \( r_1 = \frac{a}{2} \)

### Step 3: Analyze the second circle
The second circle is given by:
\[ x^2 + y^2 = c^2 \]
This is already in standard form, where:
- Center \( C_2 = (0, 0) \)
- Radius \( r_2 = c \)

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

### Step 5: Set up the touching condition
For the circles to touch each other, the distance between the centers must equal the sum or difference of the radii:
\[ d = r_1 + r_2 \quad \text{or} \quad d = |r_1 - r_2| \]
This gives us two cases to consider:
1. \( \frac{a}{2} = \frac{a}{2} + c \)
2. \( \frac{a}{2} = \left| \frac{a}{2} - c \right| \)

### Step 6: Solve the equations
**Case 1:**
\[ \frac{a}{2} = \frac{a}{2} + c \implies c = 0 \quad \text{(not valid since \( c > 0 \))} \]

**Case 2:**
\[ \frac{a}{2} = \frac{a}{2} - c \implies c = 0 \quad \text{(not valid)} \]
\[ \frac{a}{2} = c \implies a = 2c \]

### Step 7: Consider the absolute value condition
From the absolute value condition:
\[ \frac{a}{2} = c \quad \text{or} \quad \frac{a}{2} = -c \]
Since \( c > 0 \), we only consider:
\[ \frac{a}{2} = c \implies a = 2c \]

### Conclusion
Thus, the two circles touch each other if:
\[ |a| = c \]
or specifically, since \( c > 0 \):
\[ c = |a| \]