The internal centre of similitude of two circles `(x-3)^(2)+(y-2)^(2)=9,(x+5)^(2)+(y+6)^(2)=9` is

Options

A. (-1,-2)
B. (-2,-1)
C. (3,2)
D. (-5,-6)

Answer

To find the internal center of similitude of the two given circles, we can follow these steps:

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

The equations of the circles are:
1. \((x - 3)^2 + (y - 2)^2 = 9\)
2. \((x + 5)^2 + (y + 6)^2 = 9\)

From these equations, we can identify:
- For the first circle:
- Center \(C_1 = (3, 2)\)
- Radius \(R_1 = \sqrt{9} = 3\)

- For the second circle:
- Center \(C_2 = (-5, -6)\)
- Radius \(R_2 = \sqrt{9} = 3\)

### Step 2: Use the formula for the internal center of similitude

The formula for the internal center of similitude \(P\) of two circles with centers \(C_1\) and \(C_2\) and radii \(R_1\) and \(R_2\) is given by:

\[ P = \frac{R_2 C_1 + R_1 C_2}{R_1 + R_2} \]

### Step 3: Substitute the values into the formula

Substituting the values we found:
- \(C_1 = (3, 2)\)
- \(C_2 = (-5, -6)\)
- \(R_1 = 3\)
- \(R_2 = 3\)

We can calculate \(P\):

\[ P = \frac{3(3, 2) + 3(-5, -6)}{3 + 3} \]

Calculating the numerator:

\[ 3(3, 2) = (9, 6) \]
\[ 3(-5, -6) = (-15, -18) \]

Now add these two vectors:

\[ (9, 6) + (-15, -18) = (9 - 15, 6 - 18) = (-6, -12) \]

Now, divide by the sum of the radii:

\[ P = \frac{(-6, -12)}{6} = (-1, -2) \]

### Step 4: Conclusion

Thus, the internal center of similitude of the two circles is:

\[ \boxed{(-1, -2)} \]

---