There are four circles each of radius 1 unit touching both the axis. The equation of the smaller circle touching all these circle is

Options

A. `x^(2) + y ^(2) =2`
B. `x ^(2) + y ^(2) = (sqrt2 -1) ^(2)`
C. `x ^(2) + y ^(2) =4`
D. `x ^(2) + y ^(2) =1`

Answer

To find the equation of the smaller circle that touches four circles each of radius 1 unit, which are tangent to both the x-axis and y-axis, we can follow these steps:

### Step 1: Understand the positions of the four circles
The centers of the four circles can be determined based on their radius and their positions relative to the axes. The circles will be positioned at:
- Circle 1 (C1): Center at (1, 1)
- Circle 2 (C2): Center at (-1, 1)
- Circle 3 (C3): Center at (-1, -1)
- Circle 4 (C4): Center at (1, -1)

### Step 2: Determine the distance from the origin to the center of one of the circles
To find the radius of the smaller circle, we need to calculate the distance from the origin (0, 0) to the center of one of the circles, say C1 (1, 1). Using the distance formula:
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ \text{Distance} = \sqrt{(1 - 0)^2 + (1 - 0)^2} = \sqrt{1^2 + 1^2} = \sqrt{2} \]

### Step 3: Relate the radius of the smaller circle to the distance
The distance from the origin to the center of C1 is \(\sqrt{2}\). The radius of the smaller circle (r) plus the radius of the larger circles (1 unit) must equal this distance:
\[ 1 + r = \sqrt{2} \]
From this, we can solve for r:
\[ r = \sqrt{2} - 1 \]

### Step 4: Determine the center of the smaller circle
The center of the smaller circle that touches all four circles is at the origin (0, 0).

### Step 5: Write the equation of the smaller circle
The standard equation of a circle with center (h, k) and radius r is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting h = 0, k = 0, and \(r = \sqrt{2} - 1\):
\[ x^2 + y^2 = (\sqrt{2} - 1)^2 \]

### Step 6: Expand the right-hand side
Now, we need to expand \((\sqrt{2} - 1)^2\):
\[ (\sqrt{2} - 1)^2 = 2 - 2\sqrt{2} + 1 = 3 - 2\sqrt{2} \]
Thus, the equation of the smaller circle becomes:
\[ x^2 + y^2 = 3 - 2\sqrt{2} \]

### Final Answer
The equation of the smaller circle is:
\[ x^2 + y^2 = 3 - 2\sqrt{2} \]

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