A circle cuts the circles `x^(2)+y^(2)=4`
`x^(2)+y^(2)-6x-8y+10=0` and
`x^(2)+y^(2)+2x-4y-2=0`
at the ends of diameter. The co-ordinates of its centre are

Options

A. (2,3 )
B. (-2, -3)
C. (4, 6)
D. (-4, -6)

Answer

To find the coordinates of the center of the circle that cuts the given circles at the ends of the diameter, we will follow these steps:

### Step 1: Identify the equations of the circles
The equations of the circles are:
1. \( S_1: x^2 + y^2 = 4 \)
2. \( S_2: x^2 + y^2 - 6x - 8y + 10 = 0 \)
3. \( S_3: x^2 + y^2 + 2x - 4y - 2 = 0 \)

### Step 2: Rewrite the equations of circles \( S_2 \) and \( S_3 \) in standard form
For \( S_2 \):
\[ x^2 + y^2 - 6x - 8y + 10 = 0 \implies (x - 3)^2 + (y - 4)^2 = 15 \]
Thus, the center \( C_2 \) is \( (3, 4) \) and the radius \( r_2 = \sqrt{15} \).

For \( S_3 \):
\[ x^2 + y^2 + 2x - 4y - 2 = 0 \implies (x + 1)^2 + (y - 2)^2 = 5 \]
Thus, the center \( C_3 \) is \( (-1, 2) \) and the radius \( r_3 = \sqrt{5} \).

### Step 3: Set up the general equation of the circle \( S \)
Let the equation of the circle \( S \) be:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
The center of this circle is \( (-g, -f) \).

### Step 4: Use the common chord condition for circles
The common chord condition states that the circle \( S \) intersects \( S_1 \) at the ends of the diameter. Therefore, we can use the common chord formula:
\[ S - S_1 = 0 \]
This gives:
\[ 2gx + 2fy + c + 4 = 0 \]
Substituting the center of \( S_1 \) which is \( (0, 0) \):
\[ c + 4 = 0 \implies c = -4 \]

### Step 5: Apply the same condition for \( S_2 \)
Using the common chord condition for \( S_2 \):
\[ S - S_2 = 0 \]
This gives:
\[ 2gx + 2fy + c + 6x + 8y - 10 = 0 \]
Substituting the center \( (3, 4) \):
\[ 6g + 8f - 10 - 4 = 0 \implies 6g + 8f - 14 = 0 \implies 3g + 4f = 7 \quad \text{(Equation 1)} \]

### Step 6: Apply the same condition for \( S_3 \)
Using the common chord condition for \( S_3 \):
\[ S - S_3 = 0 \]
This gives:
\[ 2gx + 2fy + c - 2x + 4y + 2 = 0 \]
Substituting the center \( (-1, 2) \):
\[ -2g + 4f - 4 - 4 = 0 \implies -2g + 4f - 8 = 0 \implies -g + 2f = 4 \quad \text{(Equation 2)} \]

### Step 7: Solve the system of equations
From Equation 1:
\[ 3g + 4f = 7 \quad \text{(1)} \]
From Equation 2:
\[ -g + 2f = 4 \quad \text{(2)} \]

From (2), we can express \( g \):
\[ g = 2f - 4 \]
Substituting \( g \) into (1):
\[ 3(2f - 4) + 4f = 7 \]
\[ 6f - 12 + 4f = 7 \]
\[ 10f - 12 = 7 \implies 10f = 19 \implies f = \frac{19}{10} \]
Now substituting \( f \) back to find \( g \):
\[ g = 2\left(\frac{19}{10}\right) - 4 = \frac{38}{10} - \frac{40}{10} = -\frac{2}{10} = -\frac{1}{5} \]

### Step 8: Find the center of the circle \( S \)
The center of circle \( S \) is given by \( (-g, -f) \):
\[ \text{Center} = \left(-\left(-\frac{1}{5}\right), -\frac{19}{10}\right) = \left(\frac{1}{5}, -\frac{19}{10}\right) \]

### Final Answer
The coordinates of the center of the circle are \( \left(\frac{1}{5}, -\frac{19}{10}\right) \).