If the points `A:(0, a), B:(-2,0) and C:(1, 1)` form an obtuse angle triangle (obtuse angled at angle A), then sum of all the possible integral values of a is

Options

A. 0
B. 3
C. 2
D. 1

Answer

To find the sum of all possible integral values of \( a \) such that the points \( A(0, a) \), \( B(-2, 0) \), and \( C(1, 1) \) form an obtuse triangle with the obtuse angle at \( A \), we can follow these steps:

### Step 1: Determine the condition for an obtuse triangle
For triangle \( ABC \) to be obtuse at vertex \( A \), the following condition must hold:
\[ AB^2 + AC^2 < BC^2 \]
where \( AB \), \( AC \), and \( BC \) are the lengths of the sides opposite to vertices \( A \), \( B \), and \( C \) respectively.

### Step 2: Calculate the lengths of the sides
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: Set up the inequality
Now, substituting these lengths into the inequality:
\[ AB^2 + AC^2 < BC^2 \]
This becomes:
\[ (4 + a^2) + (a^2 - 2a + 2) < 10 \]
Simplifying this gives:
\[ 4 + a^2 + a^2 - 2a + 2 < 10 \]
\[ 2a^2 - 2a + 6 < 10 \]
\[ 2a^2 - 2a - 4 < 0 \]
Dividing the entire inequality by 2:
\[ a^2 - a - 2 < 0 \]

### Step 4: Factor the quadratic
Factoring the quadratic:
\[ (a - 2)(a + 1) < 0 \]

### Step 5: Determine the intervals
The roots of the equation \( (a - 2)(a + 1) = 0 \) are \( a = 2 \) and \( a = -1 \). The quadratic opens upwards, so the inequality \( (a - 2)(a + 1) < 0 \) holds true in the interval:
\[ -1 < a < 2 \]

### 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
Now, we sum these integral values:
\[ 0 + 1 = 1 \]

Thus, the sum of all possible integral values of \( a \) is:
\[ \boxed{1} \]