Let each of the circles,
`S_(1)=x^(2)+y^(2)+4y-1=0`,
`S_(2)=x^(2)+y^(2)+6x+y+8=0`,
`S_(3)=x^(2)+y^(2)-4x-4y-37=0`
touches the other two. Let `P_(1), P_(2), P_(3)` be the points of contact of `S_(1) and S_(2), S_(2) and S_(3), S_(3) and S_(1)` respectively and `C_(1), C_(2), C_(3)` be the centres of `S_(1), S_(2), S_(3)` respectively.
Q. The co-ordinates of `P_(1)` are :

Options

A. (2, -1)
B. (2, 1)
C. (-2, 1)
D. (-2, -1)

Answer

To solve the problem, we need to find the coordinates of the point of contact \( P_1 \) between the circles \( S_1 \) and \( S_2 \). We will follow these steps:

### Step 1: Find the center and radius of circle \( S_1 \)

The equation of circle \( S_1 \) is given by:
\[ S_1: x^2 + y^2 + 4y - 1 = 0 \]
To find the center and radius, we rewrite this in standard form by completing the square.

1. Rearrange the equation:
\[ x^2 + (y^2 + 4y) = 1 \]
2. Complete the square for \( y \):
\[ y^2 + 4y = (y + 2)^2 - 4 \]
Thus, we have:
\[ x^2 + (y + 2)^2 - 4 = 1 \implies x^2 + (y + 2)^2 = 5 \]
3. From this, we can identify the center \( C_1 \) and radius \( R_1 \):
- Center \( C_1 = (0, -2) \)
- Radius \( R_1 = \sqrt{5} \)

### Step 2: Find the center and radius of circle \( S_2 \)

The equation of circle \( S_2 \) is:
\[ S_2: x^2 + y^2 + 6x + y + 8 = 0 \]
1. Rearrange the equation:
\[ (x^2 + 6x) + (y^2 + y) = -8 \]
2. Complete the square for \( x \) and \( y \):
- For \( x \):
\[ x^2 + 6x = (x + 3)^2 - 9 \]
- For \( y \):
\[ y^2 + y = (y + \frac{1}{2})^2 - \frac{1}{4} \]
3. Substitute back:
\[ (x + 3)^2 - 9 + (y + \frac{1}{2})^2 - \frac{1}{4} = -8 \]
Simplifying gives:
\[ (x + 3)^2 + (y + \frac{1}{2})^2 = \frac{9}{4} \]
4. From this, we can identify the center \( C_2 \) and radius \( R_2 \):
- Center \( C_2 = (-3, -\frac{1}{2}) \)
- Radius \( R_2 = \frac{3}{2} \)

### Step 3: Find the point of contact \( P_1 \)

Since circles \( S_1 \) and \( S_2 \) touch each other, the distance between their centers \( C_1 \) and \( C_2 \) must equal the sum of their radii.

1. Calculate the distance \( C_1 C_2 \):
\[ C_1 C_2 = \sqrt{(0 - (-3))^2 + (-2 - (-\frac{1}{2}))^2} = \sqrt{3^2 + (-2 + \frac{1}{2})^2} \]
Simplifying:
\[ = \sqrt{9 + (-\frac{3}{2})^2} = \sqrt{9 + \frac{9}{4}} = \sqrt{\frac{36}{4} + \frac{9}{4}} = \sqrt{\frac{45}{4}} = \frac{3\sqrt{5}}{2} \]
2. The sum of the radii:
\[ R_1 + R_2 = \sqrt{5} + \frac{3}{2} \]
To compare, we can convert \( \sqrt{5} \) to a decimal or a fraction, but we will use the ratio for finding \( P_1 \).

3. The ratio of the distances from \( C_1 \) to \( P_1 \) and from \( C_2 \) to \( P_1 \) is given by the ratio of the radii:
\[ \text{Ratio} = R_1 : R_2 = \sqrt{5} : \frac{3}{2} = 2 : 1 \]

Using the section formula to find \( P_1 \):
\[ P_1 = \left( \frac{2(-3) + 1(0)}{2 + 1}, \frac{2(-\frac{1}{2}) + 1(-2)}{2 + 1} \right) \]
Calculating:
\[ P_1 = \left( \frac{-6}{3}, \frac{-1 - 2}{3} \right) = \left( -2, -1 \right) \]

### Final Answer:
The coordinates of \( P_1 \) are \( (-2, -1) \).

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