Question

The points A(a, 2), B(3, 1), C(-1, 3) do not form a triangle if a=

• A. 2
• B. 3
• C. -2
• D. 1

Answer 1

The Correct Option is D

The points do not form a triangle when they are collinear. To check for collinearity, we can use the determinant formula. For three points \( A(x_1, y_1), B(x_2, y_2), C(x_3, y_3) \), they are collinear if the following determinant is zero: \[ \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} = 0 \] Substitute the coordinates: \[ \begin{vmatrix} a & 2 & 1 \\ 3 & 1 & 1 \\ -1 & 3 & 1 \end{vmatrix} = 0 \] Expanding the determinant: \[ a(1 \cdot 1 - 1 \cdot 3) - 2(3 \cdot 1 - (-1) \cdot 1) + 1(3 \cdot 3 - (-1) \cdot 1) \] \[ = a(1 - 3) - 2(3 + 1) + (9 + 1) \] \[ = a(-2) - 2(4) + 10 = 0 \] \[ -2a - 8 + 10 = 0 \Rightarrow -2a + 2 = 0 \Rightarrow a = 1 \]