Question

What is the value of '$k$' for which the points $(-1, 2)$, $(1, 4)$ and $(3, k)$ are collinear?

• A. $0$
• B. $2$
• C. $4$
• D. $6$

Hint:

For collinearity, always equate slopes or use the area method. Slope method is faster when only one variable is unknown.

Answer 1

The Correct Option is D

Concept: Three points are collinear if the slope between any two pairs of points is the same. Slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] If points $A, B, C$ are collinear: \[ m_{AB} = m_{BC} \]

Step 1: Given points \[ A(-1,2), \quad B(1,4), \quad C(3,k) \]

Step 2: Slope of AB \[ m_{AB} = \frac{4-2}{1-(-1)} = \frac{2}{2} = 1 \]

Step 3: Slope of BC \[ m_{BC} = \frac{k-4}{3-1} = \frac{k-4}{2} \]

Step 4: Equate slopes \[ 1 = \frac{k-4}{2} \]

Step 5: Solve \[ 2 = k - 4 \] \[ k = 6 \]