Consider two circes `x ^(2) + y ^(2) =a ^(2) - lamda and x ^(2) + y ^(2) - 2 ax cos theta - 2 a y sin theta + lamda = 0.` Each circle passes through the centre of the other for

Options

A. all values of `lamda `
B. no vlaue of `lamda`
C. only one value of `lamda`
D. more than one vlaue of `lamda`

Answer

To find the value of \(\lambda\) for which each circle passes through the center of the other, we will analyze the given equations of the circles step by step.

### Step 1: Identify the equations of the circles
The equations of the circles are given as:
1. \(x^2 + y^2 = a^2 - \lambda\) (Circle 1)
2. \(x^2 + y^2 - 2ax \cos \theta - 2ay \sin \theta + \lambda = 0\) (Circle 2)

### Step 2: Rewrite the equations in standard form
For Circle 1, we can rewrite it as:
\[ x^2 + y^2 + 0 \cdot x + 0 \cdot y + (\lambda - a^2) = 0 \]
This indicates that the center of Circle 1 is at \((0, 0)\).

For Circle 2, we can rearrange it as:
\[ x^2 + y^2 - 2ax \cos \theta - 2ay \sin \theta + \lambda = 0 \]
This can be rewritten to identify the center:
\[ x^2 + y^2 + 2\left(-a \cos \theta\right)x + 2\left(-a \sin \theta\right)y + \lambda = 0 \]
Thus, the center of Circle 2 is at \((a \cos \theta, a \sin \theta)\).

### Step 3: Condition for Circle 1 to pass through the center of Circle 2
For Circle 1 to pass through the center of Circle 2, the coordinates \((a \cos \theta, a \sin \theta)\) must satisfy the equation of Circle 1:
\[ (a \cos \theta)^2 + (a \sin \theta)^2 = a^2 - \lambda \]
Calculating the left-hand side:
\[ a^2 (\cos^2 \theta + \sin^2 \theta) = a^2 \cdot 1 = a^2 \]
Thus, we have:
\[ a^2 = a^2 - \lambda \]
This simplifies to:
\[ \lambda = 0 \]

### Step 4: Condition for Circle 2 to pass through the center of Circle 1
Now, we check if Circle 2 passes through the center of Circle 1, which is at \((0, 0)\):
Substituting \(x = 0\) and \(y = 0\) into Circle 2's equation:
\[ 0^2 + 0^2 - 2a(0) \cos \theta - 2a(0) \sin \theta + \lambda = 0 \]
This simplifies to:
\[ \lambda = 0 \]

### Conclusion
Both conditions yield \(\lambda = 0\). Therefore, the only value of \(\lambda\) for which each circle passes through the center of the other is:
\[ \lambda = 0 \]