To solve the problem, we need to determine the conditions under which the triangle formed by the points A(0, a), B(-2, 0), and C(1, 1) has an obtuse angle at point A.
### Step 1: Determine the coordinates of points B and C
The coordinates of points B and C are given as:
- B: (-2, 0)
- C: (1, 1)
### Step 2: Find the lengths of the sides of the triangle
We will calculate the lengths of the sides of the triangle using the distance formula. The length of a side between two points (x1, y1) and (x2, y2) is given by:
\[ d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \]
1. Length AB:
\[
AB = \sqrt{(0 - (-2))^2 + (a - 0)^2} = \sqrt{(2)^2 + a^2} = \sqrt{4 + a^2}
\]
2. Length AC:
\[
AC = \sqrt{(0 - 1)^2 + (a - 1)^2} = \sqrt{(-1)^2 + (a - 1)^2} = \sqrt{1 + (a - 1)^2} = \sqrt{1 + a^2 - 2a + 1} = \sqrt{a^2 - 2a + 2}
\]
3. Length BC:
\[
BC = \sqrt{(-2 - 1)^2 + (0 - 1)^2} = \sqrt{(-3)^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10}
\]
### Step 3: Apply the condition for an obtuse triangle
For triangle ABC to have an obtuse angle at A, the following condition must hold:
\[
AB^2 + AC^2 < BC^2
\]
Substituting the lengths we found:
\[
(4 + a^2) + (a^2 - 2a + 2) < 10
\]
Simplifying this:
\[
4 + a^2 + a^2 - 2a + 2 < 10
\]
\[
2a^2 - 2a + 6 < 10
\]
\[
2a^2 - 2a - 4 < 0
\]
Dividing the entire inequality by 2:
\[
a^2 - a - 2 < 0
\]
### Step 4: Factor the quadratic inequality
Factoring the quadratic:
\[
(a - 2)(a + 1) < 0
\]
### Step 5: Determine the intervals
To find the intervals where the product is negative, we find the roots:
- \( a - 2 = 0 \Rightarrow a = 2 \)
- \( a + 1 = 0 \Rightarrow a = -1 \)
The critical points divide the number line into intervals:
1. \( (-\infty, -1) \)
2. \( (-1, 2) \)
3. \( (2, \infty) \)
Testing intervals:
- For \( a < -1 \) (e.g., \( a = -2 \)): \( (-)(-) > 0 \) (not valid)
- For \( -1 < a < 2 \) (e.g., \( a = 0 \)): \( (+)(-) < 0 \) (valid)
- For \( a > 2 \) (e.g., \( a = 3 \)): \( (+)(+) > 0 \) (not valid)
Thus, the solution to the inequality is:
\[
-1 < a < 2
\]
### Step 6: Identify integral values of \( a \)
The integral values of \( a \) in the interval \( (-1, 2) \) are:
- \( 0 \)
- \( 1 \)
### Step 7: Calculate the sum of integral values
The sum of all possible integral values of \( a \) is:
\[
0 + 1 = 1
\]
### Final Answer
The sum of all possible integral values of \( a \) is **1**.
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