Let each of the circles
`S_(1)-=x^(2)+y^(2)+4y-1=0`
`S_(1)-= x^(2)+y^(2)+6x+y+8=0`
`S_(3)-=x^(2)+y^(2)-4x-4y-37=0`
touch the other two. Also, let `P_(1),P_(2)` and `P_(3)` be the points of contact of `S_(1)` and `S_(2) , S_(2)` and `S_(3)`, and `S_(3)` , respectively, `C_(1),C_(2)` and `C_(3)` are the centres of `S_(1),S_(2)` and `S_(3)` respectively.
The ratio `("area"(DeltaP_(1)P_(2)P_(3)))/("area"(DeltaC_(1)C_(2)C_(3)))` is equal to

Options

A. `3:2`
B. `2:3`
C. `5:3`
D. `2:5`

Explanation

To solve the problem, we need to find the ratio of the areas of two triangles: one formed by the points of contact of the circles and the other formed by the centers of the circles. Let's break down the solution step by step.

### Step 1: Find the centers and radii of the circles

1. **Circle \( S_1 \)**:
\[ x^2 + y^2 + 4y - 1 = 0 \implies x^2 + (y + 2)^2 = 5 \]
- Center \( C_1 = (0, -2) \)
- Radius \( r_1 = \sqrt{5} \)

2. **Circle \( S_2 \)**:
\[ x^2 + y^2 + 6x + y + 8 = 0 \implies (x + 3)^2 + (y + \frac{1}{2})^2 = \frac{65}{4} \]
- Center \( C_2 = (-3, -\frac{1}{2}) \)
- Radius \( r_2 = \frac{\sqrt{65}}{2} \)

3. **Circle \( S_3 \)**:
\[ x^2 + y^2 - 4x - 4y - 37 = 0 \implies (x - 2)^2 + (y - 2)^2 = 41 \]
- Center \( C_3 = (2, 2) \)
- Radius \( r_3 = \sqrt{41} \)

### Step 2: Find the distances between the centers

1. **Distance \( C_1C_2 \)**:
\[ d_{C_1C_2} = \sqrt{(-3 - 0)^2 + \left(-\frac{1}{2} + 2\right)^2} = \sqrt{9 + \left(\frac{3}{2}\right)^2} = \sqrt{9 + \frac{9}{4}} = \sqrt{\frac{45}{4}} = \frac{3\sqrt{5}}{2} \]

2. **Distance \( C_2C_3 \)**:
\[ d_{C_2C_3} = \sqrt{(2 + 3)^2 + (2 + \frac{1}{2})^2} = \sqrt{25 + \left(\frac{5}{2}\right)^2} = \sqrt{25 + \frac{25}{4}} = \sqrt{\frac{125}{4}} = \frac{5\sqrt{5}}{2} \]

3. **Distance \( C_1C_3 \)**:
\[ d_{C_1C_3} = \sqrt{(2 - 0)^2 + (2 + 2)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \]

### Step 3: Find the points of contact

1. **Point \( P_1 \)** (between \( S_1 \) and \( S_2 \)):
- Divides \( C_1C_2 \) in the ratio \( r_1 : r_2 \):
\[ P_1 = \left( \frac{r_2 \cdot 0 + r_1 \cdot (-3)}{r_1 + r_2}, \frac{r_2 \cdot (-2) + r_1 \cdot (-\frac{1}{2})}{r_1 + r_2} \right) \]

2. **Point \( P_2 \)** (between \( S_2 \) and \( S_3 \)):
- Divides \( C_2C_3 \) in the ratio \( r_2 : r_3 \):
\[ P_2 = \left( \frac{r_3 \cdot (-3) + r_2 \cdot 2}{r_2 + r_3}, \frac{r_3 \cdot (-\frac{1}{2}) + r_2 \cdot 2}{r_2 + r_3} \right) \]

3. **Point \( P_3 \)** (between \( S_3 \) and \( S_1 \)):
- Divides \( C_1C_3 \) in the ratio \( r_3 : r_1 \):
\[ P_3 = \left( \frac{r_1 \cdot 2 + r_3 \cdot 0}{r_1 + r_3}, \frac{r_1 \cdot 2 + r_3 \cdot (-2)}{r_1 + r_3} \right) \]

### Step 4: Calculate the areas of the triangles

1. **Area of triangle \( P_1P_2P_3 \)**:
Using the determinant formula for the area of a triangle formed by points \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \):
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

2. **Area of triangle \( C_1C_2C_3 \)**:
Similarly, apply the formula using the coordinates of the centers.

### Step 5: Find the ratio of the areas

Finally, compute the ratio:
\[ \text{Ratio} = \frac{\text{Area of } P_1P_2P_3}{\text{Area of } C_1C_2C_3} \]