If the points A(-3, 9), B(a, b) and C(4, -5) are collinear and a + b = 1, then the values of a and b are :

Question

Accepted Answer

To determine why Option 2 is the correct answer, we need to verify the collinearity of points A(-3, 9), B(a, b), and C(4, -5) and ensure that a + b = 1.

### Explanation for Option 2 (Correct Answer: a = 2, b = -1):

1. **Collinearity Condition**: For three points to be collinear, the area of the triangle formed by them must be zero. The formula for the area of a triangle given points $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is:
   $$
   \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|
   $$
   Setting the area to zero for collinearity:
   $$
   -3(b + 5) + a(-5 - 9) + 4(9 - b) = 0
   $$
   Simplifying:
   $$
   -3b - 15 - 14a + 36 - 4b = 0
   $$
   $$
   -7b - 14a + 21 = 0
   $$
   $$
   14a + 7b = 21
   $$
   $$
   2a + b = 3 \quad \text{(Equation 1)}
   $$

2. **Given Condition**: We are also given that:
   $$
   a + b = 1 \quad \text{(Equation 2)}
   $$

3. **Solving the Equations**:
   - From Equation 2: $b = 1 - a$.
   - Substitute $b = 1 - a$ into Equation 1:
     $$
     2a + (1 - a) = 3
     $$
     $$
     2a + 1 - a = 3
     $$
     $$
     a + 1 = 3
     $$
     $$
     a = 2
     $$
   - Substitute $a = 2$ back into Equation 2:
     $$
     2 + b = 1
     $$
     $$
     b = -1
     $$

Thus, the values $a = 2$ and $b = -1$ satisfy both the collinearity condition and the given condition $a + b = 1$. Therefore, Option 2 is correct.

### Explanation for Why Other Options are Incorrect:

1. **Option 1 (a = 2, b = 1)**:
   - Substituting into $a + b = 1$: $2 + 1 = 3$, which does not satisfy the condition $a + b = 1$.

2. **Option 3 (a = -2, b = 1)**:
   - Substituting into $a + b = 1$: $-2 + 1 = -1$, which does not satisfy the condition $a + b = 1$.

3. **Option 4 (a = -2, b = -1)**:
   - Substituting into $a + b = 1$: $-2 - 1 = -3$, which does not satisfy the condition $a + b = 1$.

In conclusion, only Option 2 satisfies both the collinearity condition and the given condition $a + b = 1$.