The external centre of similitude of the circle `x^(2)+y^(2)-12x+8y+48=0` and `x^(2)+y^(2)-4x+2y-4=0` divides the segment joining centres in the ratio.

Options

A. 0.085416666666667
B. 0.12638888888889
C. `-2:3`
D. `-3:2`

Answer

To solve the problem, we need to find the external center of similitude of the two given circles and determine the ratio in which it divides the segment joining their centers.

### Step 1: Identify the equations of the circles
The equations of the circles are:
1. Circle 1: \( x^2 + y^2 - 12x + 8y + 48 = 0 \)
2. Circle 2: \( x^2 + y^2 - 4x + 2y - 4 = 0 \)

### Step 2: Find the centers and radii of the circles
For Circle 1:
- The general form of the circle is \( (x - h)^2 + (y - k)^2 = r^2 \).
- Rearranging the equation, we have:
\[ (x^2 - 12x) + (y^2 + 8y) + 48 = 0 \]
Completing the square:
\[ (x^2 - 12x + 36) + (y^2 + 8y + 16) = 4 \]
This gives us:
\[ (x - 6)^2 + (y + 4)^2 = 4 \]
- Therefore, the center \( C_1 \) is \( (6, -4) \) and the radius \( r_1 = 2 \).

For Circle 2:
- Rearranging the equation:
\[ (x^2 - 4x) + (y^2 + 2y) - 4 = 0 \]
Completing the square:
\[ (x^2 - 4x + 4) + (y^2 + 2y + 1) = 9 \]
This gives us:
\[ (x - 2)^2 + (y + 1)^2 = 9 \]
- Therefore, the center \( C_2 \) is \( (2, -1) \) and the radius \( r_2 = 3 \).

### Step 3: Find the ratio of the radii
The external center of similitude divides the segment joining the centers \( C_1 \) and \( C_2 \) in the ratio of the radii of the circles. The ratio is given by:
\[ \frac{r_1}{r_2} = \frac{2}{3} \]

### Step 4: Determine the external division
Since we are looking for the external center of similitude, the ratio will be negative:
\[ \text{Ratio} = -\frac{r_1}{r_2} = -\frac{2}{3} \]

### Conclusion
Thus, the external center of similitude divides the segment joining the centers in the ratio \( -2:3 \).