The circle `x^(2)+y^(2)-2x+4y+4=0,x^(2)+y^(2)+4x-2y+1=0` are

Options

A. intersecting
B. touching externally
C. touching internally
D. one is lying inside the other

Answer

To determine the relationship between the two given circles, we will follow these steps:

### Step 1: Write the equations of the circles in standard form.
The equations of the circles are given as:
1. \( S_1: x^2 + y^2 - 2x + 4y + 4 = 0 \)
2. \( S_2: x^2 + y^2 + 4x - 2y + 1 = 0 \)

### Step 2: Identify the center and radius of the first circle \( S_1 \).
We can rewrite the equation \( S_1 \) in the standard form \( (x - h)^2 + (y - k)^2 = r^2 \).

1. Rearranging \( S_1 \):
\[ x^2 - 2x + y^2 + 4y + 4 = 0 \]
Completing the square for \( x \) and \( y \):
\[ (x - 1)^2 - 1 + (y + 2)^2 - 4 + 4 = 0 \]
This simplifies to:
\[ (x - 1)^2 + (y + 2)^2 = 1 \]
From this, we can see that the center \( C_1 \) is \( (1, -2) \) and the radius \( r_1 = 1 \).

### Step 3: Identify the center and radius of the second circle \( S_2 \).
Now we will rewrite the equation \( S_2 \):
1. Rearranging \( S_2 \):
\[ x^2 + 4x + y^2 - 2y + 1 = 0 \]
Completing the square for \( x \) and \( y \):
\[ (x + 2)^2 - 4 + (y - 1)^2 - 1 + 1 = 0 \]
This simplifies to:
\[ (x + 2)^2 + (y - 1)^2 = 4 \]
From this, we can see that the center \( C_2 \) is \( (-2, 1) \) and the radius \( r_2 = 2 \).

### Step 4: Calculate the distance between the centers \( C_1 \) and \( C_2 \).
Using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of \( C_1(1, -2) \) and \( C_2(-2, 1) \):
\[ d = \sqrt{((-2) - 1)^2 + (1 - (-2))^2} = \sqrt{(-3)^2 + (3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]

### Step 5: Compare the distance with the sum of the radii.
Now we compare the distance \( d \) with the sum of the radii \( r_1 + r_2 \):
\[ r_1 + r_2 = 1 + 2 = 3 \]
Since \( d = 3\sqrt{2} \) and \( 3\sqrt{2} \) is greater than \( 3 \), this implies that the circles do not intersect and one circle lies inside the other.

### Conclusion:
The circles are such that one is lying inside the other.