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:

For three points to be collinear, the slopes between any two pairs of points must be the same.

Answer 1

The Correct Option is B

For three points to be collinear, the slope between any two pairs of points must be the same. The slope between points \( (1, 2) \) and \( (0, 0) \) is: \[ \text{Slope} = \frac{2 - 0}{1 - 0} = 2. \] The slope between points \( (0, 0) \) and \( (a, b) \) is: \[ \text{Slope} = \frac{b - 0}{a - 0} = \frac{b}{a}. \] For the points to be collinear, these slopes must be equal: \[ \frac{b}{a} = 2 \quad \Rightarrow \quad a = 2b. \] Thus, the correct answer is \( \boxed{a = 2b} \).