Let A (-2, 0) and B(2, 0), then the number of integral values of a, `a in [-10, 10] for which line segment AB subtends an acute angle at point C (a, a+1) is

Options

A. 15
B. 17
C. 19
D. 21

Answer

To solve the problem, we need to determine how many integral values of \( a \) in the interval \([-10, 10]\) allow the line segment \( AB \) to subtend an acute angle at the point \( C(a, a+1) \).

### Step-by-Step Solution:

1. **Identify Points A and B**:
- Let \( A = (-2, 0) \) and \( B = (2, 0) \).

2. **Determine Coordinates of Point C**:
- The coordinates of point \( C \) are given as \( C(a, a+1) \).

3. **Calculate Distances**:
- The distance \( AC \) can be calculated using the distance formula:
\[ AC = \sqrt{(a - (-2))^2 + ((a + 1) - 0)^2} = \sqrt{(a + 2)^2 + (a + 1)^2} \]
- The distance \( BC \) is:
\[ BC = \sqrt{(a - 2)^2 + ((a + 1) - 0)^2} = \sqrt{(a - 2)^2 + (a + 1)^2} \]
- The distance \( AB \) is:
\[ AB = \sqrt{((-2) - 2)^2 + (0 - 0)^2} = \sqrt{(-4)^2} = 4 \]

4. **Condition for Acute Angle**:
- For the angle at \( C \) to be acute, the following condition must hold:
\[ AC^2 + BC^2 > AB^2 \]
- Since \( AB = 4 \), we have \( AB^2 = 16 \).

5. **Calculate \( AC^2 \) and \( BC^2 \)**:
- Calculate \( AC^2 \):
\[ AC^2 = (a + 2)^2 + (a + 1)^2 = (a^2 + 4a + 4) + (a^2 + 2a + 1) = 2a^2 + 6a + 5 \]
- Calculate \( BC^2 \):
\[ BC^2 = (a - 2)^2 + (a + 1)^2 = (a^2 - 4a + 4) + (a^2 + 2a + 1) = 2a^2 - 2a + 5 \]

6. **Combine and Set Inequality**:
- Combine \( AC^2 \) and \( BC^2 \):
\[ AC^2 + BC^2 = (2a^2 + 6a + 5) + (2a^2 - 2a + 5) = 4a^2 + 4a + 10 \]
- Set up the inequality:
\[ 4a^2 + 4a + 10 > 16 \]
- Simplifying gives:
\[ 4a^2 + 4a - 6 > 0 \quad \Rightarrow \quad 2a^2 + 2a - 3 > 0 \]

7. **Solve the Quadratic Inequality**:
- Factor the quadratic:
\[ 2a^2 + 2a - 3 = 0 \quad \Rightarrow \quad a^2 + a - \frac{3}{2} = 0 \]
- Using the quadratic formula:
\[ a = \frac{-1 \pm \sqrt{1 + 6}}{2} = \frac{-1 \pm \sqrt{7}}{2} \]
- The roots are approximately \( a \approx 1.822 \) and \( a \approx -2.822 \).

8. **Determine the Intervals**:
- The inequality \( 2a^2 + 2a - 3 > 0 \) holds outside the roots:
- \( a < -2.822 \) or \( a > 1.822 \).

9. **Count Integral Values**:
- The integral values of \( a \) in the interval \([-10, 10]\):
- From \( -10 \) to \( -3 \): \( -10, -9, -8, -7, -6, -5, -4, -3 \) (8 values)
- From \( 2 \) to \( 10 \): \( 2, 3, 4, 5, 6, 7, 8, 9, 10 \) (9 values)

10. **Total Integral Values**:
- Total integral values = \( 8 + 9 = 17 \).

### Final Answer:
The number of integral values of \( a \) for which the line segment \( AB \) subtends an acute angle at point \( C \) is **17**.