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, we need to analyze the two circles given by the equations:

1. \( x^2 + y^2 = ax \)
2. \( x^2 + y^2 = c^2 \) (where \( c > 0 \))

### Step 1: Rewrite the equations in standard form

The first equation can be rewritten as:
\[ x^2 - ax + y^2 = 0 \]
Completing the square for the \( x \) terms:
\[ (x - \frac{a}{2})^2 + y^2 = \frac{a^2}{4} \]
This shows that the center of the first circle \( C_1 \) is \( \left(\frac{a}{2}, 0\right) \) and its radius \( r_1 \) is \( \frac{a}{2} \).

The second equation is already in standard form:
\[ x^2 + y^2 = c^2 \]
This shows that the center of the second circle \( C_2 \) is \( (0, 0) \) and its radius \( r_2 \) is \( c \).

### Step 2: Find the distance between the centers of the circles

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 3: Use the condition for the circles to touch each other

For two circles to touch each other externally, the distance between their centers must equal the sum of their radii:
\[ d = r_1 + r_2 \]
Substituting the values we found:
\[ \frac{a}{2} = \frac{a}{2} + c \]

### Step 4: Rearranging the equation

Rearranging gives:
\[ \frac{a}{2} - \frac{a}{2} = c \]
This simplifies to:
\[ 0 = c \]
This is not possible since \( c > 0 \). Therefore, we need to consider the case when the circles touch each other internally, which gives us:
\[ d = |r_1 - r_2| \]
Substituting the values:
\[ \frac{a}{2} = \left| \frac{a}{2} - c \right| \]

### Step 5: Analyze the absolute value condition

This leads to two cases:

**Case 1:**
\[ \frac{a}{2} = \frac{a}{2} - c \implies c = 0 \quad (\text{not valid since } c > 0) \]

**Case 2:**
\[ \frac{a}{2} = c - \frac{a}{2} \implies c = a \]

### Step 6: Find the ratio \( \left| \frac{c}{a} \right| \)

From \( c = a \), we have:
\[ \left| \frac{c}{a} \right| = \left| \frac{a}{a} \right| = 1 \]

### Conclusion

Thus, the value of \( \left| \frac{c}{a} \right| \) is \( 1 \).

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