To solve the problem step by step, we will follow the instructions given in the video transcript and derive the necessary values.
### Step 1: Understand the Given Points
We have the point \( A(1, 1) \) on the ellipse and the point \( B(7, -6) \) where the tangent at point \( A \) intersects the directrix of the ellipse. We need to find the coordinates of point \( C(\alpha, \beta) \), which is the center of triangle \( \triangle SAB \) where \( S \) is the focus of the ellipse.
### Step 2: Determine the Midpoint \( C \)
The coordinates of point \( C \) can be found as the midpoint of segment \( AB \). The formula for the midpoint \( C(x, y) \) of two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by:
\[
C\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
\]
Substituting the coordinates of points \( A \) and \( B \):
- \( A(1, 1) \)
- \( B(7, -6) \)
Calculating the coordinates of \( C \):
\[
C\left(\frac{1 + 7}{2}, \frac{1 - 6}{2}\right) = C\left(\frac{8}{2}, \frac{-5}{2}\right) = C(4, -\frac{5}{2})
\]
### Step 3: Identify \( \alpha \) and \( \beta \)
From the coordinates of point \( C \):
- \( \alpha = 4 \)
- \( \beta = -\frac{5}{2} \)
### Step 4: Calculate \( \alpha - \beta \) and \( \alpha + \beta \)
Now we will calculate \( \alpha - \beta \) and \( \alpha + \beta \):
1. **Calculate \( \alpha - \beta \)**:
\[
\alpha - \beta = 4 - \left(-\frac{5}{2}\right) = 4 + \frac{5}{2} = \frac{8}{2} + \frac{5}{2} = \frac{13}{2}
\]
2. **Calculate \( \alpha + \beta \)**:
\[
\alpha + \beta = 4 + \left(-\frac{5}{2}\right) = 4 - \frac{5}{2} = \frac{8}{2} - \frac{5}{2} = \frac{3}{2}
\]
### Step 5: Calculate the Distance \( SC \)
To find the distance \( SC \), we need to know the coordinates of the focus \( S \). However, since we don't have the coordinates of \( S \) directly, we can express the distance \( SC \) in terms of the distance \( AB \):
\[
SC = \frac{1}{2} AB
\]
Where \( AB \) is calculated as follows:
\[
AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(7 - 1)^2 + (-6 - 1)^2} = \sqrt{(6)^2 + (-7)^2} = \sqrt{36 + 49} = \sqrt{85}
\]
Thus,
\[
SC = \frac{1}{2} \sqrt{85}
\]
### Final Results
- \( C(4, -\frac{5}{2}) \)
- \( \alpha - \beta = \frac{13}{2} \)
- \( \alpha + \beta = \frac{3}{2} \)
- \( SC = \frac{1}{2} \sqrt{85} \)
