Question

If the points \( (1, 2) \), \( (0, 0) \), and \( (a, b) \) are collinear, then:

• A. \( a = b \)
• B. \( a = 2b \)
• C. \( 2a = b \)
• D. \( a + b = 0 \)

Hint:

Three points are collinear if the area of the triangle they form is zero.

Answer 1

The Correct Option is C

Three points are collinear if the area of the triangle formed by them is zero.
Using the determinant formula:
\[ \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| = 0. \] Substituting values:
\[ \frac{1}{2} \left| 1(0 - b) + 0(b - 2) + a(2 - 0) \right| = 0. \] \[ \left| -b + 2a \right| = 0. \] \[ b = 2a. \]