If circles `x^(2)+y^(2)+2x+2y+c=0` and `x^(2)+y^(2)+2ax+2ay+c=0` where `c in R^(+), a != 1` are such that one circle lies inside the other, then

Options

A. `a in (0, sqrt((c)/(2)))-{1}`
B. `a in (- sqrt((c)/(2)),sqrt((c)/(2)))-{1}`
C. ` a in (-sqrt((c)/(2)),0)`
D. none of these

Explanation

To solve the problem, we need to analyze the given circles and determine the conditions under which one circle lies inside the other.

### Step 1: Write the equations of the circles
The equations of the circles are:
1. Circle 1: \( x^2 + y^2 + 2x + 2y + c = 0 \)
2. Circle 2: \( x^2 + y^2 + 2ax + 2ay + c = 0 \)

### Step 2: Rewrite the equations in standard form
We can rewrite both equations in the standard form of a circle \((x - h)^2 + (y - k)^2 = r^2\).

For Circle 1:
\[ x^2 + y^2 + 2x + 2y + c = 0 \implies (x + 1)^2 + (y + 1)^2 = 1 - c \]
Thus, the center \(C_1\) is \((-1, -1)\) and radius \(r_1 = \sqrt{1 - c}\).

For Circle 2:
\[ x^2 + y^2 + 2ax + 2ay + c = 0 \implies (x + a)^2 + (y + a)^2 = 1 - c \]
Thus, the center \(C_2\) is \((-a, -a)\) and radius \(r_2 = \sqrt{1 - c}\).

### Step 3: Determine the conditions for one circle to lie inside the other
For one circle to lie inside the other, the following conditions must be satisfied:
1. The distance between the centers must be less than the difference of the radii.
2. The circles must be on the same side of the radical axis.

### Step 4: Find the distance between the centers
The distance \(d\) between the centers \(C_1\) and \(C_2\) is given by:
\[ d = \sqrt{((-1) - (-a))^2 + ((-1) - (-a))^2} = \sqrt{(a - 1)^2 + (a - 1)^2} = \sqrt{2(a - 1)^2} = \sqrt{2} |a - 1| \]

### Step 5: Set up the inequality for one circle to lie inside the other
For Circle 1 to lie inside Circle 2:
\[ d < r_2 - r_1 \]
This leads to:
\[ \sqrt{2} |a - 1| < \sqrt{1 - c} - \sqrt{1 - c} \]
This simplifies to:
\[ \sqrt{2} |a - 1| < 0 \]
This is not possible unless \(c\) is such that the circles do not intersect.

### Step 6: Analyze the conditions
Since \(c\) is a positive real number, we can conclude that:
1. \(a > 0\) (from the product of the centers being positive).
2. The circles must not intersect, which leads to \(c < 1\).

### Conclusion
Thus, the conditions for one circle to lie inside the other are:
- \(c < 1\)
- \(a > 0\)