Finding the Internal Center of Similitude for Two Circles

This solution explains how to find the internal center of similitude for the two given circles. The internal center of similitude is a specific point related to two circles, crucial in geometry.

Analyzing the Given Circles

First, let's identify the centers and radii of the two circles provided:

• Circle 1: $ (x - 1)^2 + (y - 3)^2 = 4 $
  • Center $ C_1 $ is at $ (x_1, y_1) = (1, 3) $.
  • Radius $ r_1 $ is $ \sqrt{4} = 2 $.
• Circle 2: $ (x + 5)^2 + (y - 9)^2 = 16 $
  • Center $ C_2 $ is at $ (x_2, y_2) = (-5, 9) $.
  • Radius $ r_2 $ is $ \sqrt{16} = 4 $.

Understanding the Internal Center of Similitude

The internal center of similitude (also known as the internal homothetic center) is a point that divides the line segment connecting the centers of the two circles internally in the ratio of their radii. Let the two circles have centers $ C_1(x_1, y_1) $ and $ C_2(x_2, y_2) $ and radii $ r_1 $ and $ r_2 $, respectively. The coordinates of the internal center of similitude, T, are calculated using the section formula for internal division.

Formula for Internal Center of Similitude

The formula to calculate the coordinates $ (x, y) $ of the internal center of similitude is:

$ T(x, y) = \left( \frac{r_2 x_1 + r_1 x_2}{r_1 + r_2}, \frac{r_2 y_1 + r_1 y_2}{r_1 + r_2} \right) $

Step-by-Step Calculation

Now, let's substitute the values from our circles into the formula:

• $ x_1 = 1, y_1 = 3, r_1 = 2 $
• $ x_2 = -5, y_2 = 9, r_2 = 4 $

Calculate the x-coordinate:

$ x = \frac{(4)(1) + (2)(-5)}{2 + 4} $

$ x = \frac{4 - 10}{6} $

$ x = \frac{-6}{6} $

$ x = -1 $

Calculate the y-coordinate:

$ y = \frac{(4)(3) + (2)(9)}{2 + 4} $

$ y = \frac{12 + 18}{6} $

$ y = \frac{30}{6} $

$ y = 5 $

Conclusion

Therefore, the coordinates of the internal center of similitude for the given two circles are $ (-1, 5) $. This point lies on the line segment connecting the centers $ (1, 3) $ and $ (-5, 9) $.