Question

For what value of k points (1, 1), (3, k) and (-1, 4) are collinear?

Hint:

Using the area of a triangle formula for collinearity (\(Area = 0\)) is also a reliable method. The formula is \(\frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)| = 0\). It avoids division and can sometimes be quicker.

Answer 1

Step 1: Understanding the Concept:
If three points are collinear, they lie on the same straight line. This means the slope between any two pairs of these points must be the same. Another method is to use the fact that the area of a triangle formed by three collinear points is zero. We will use the slope method.

Step 2: Key Formula or Approach:
Let the points be A(1, 1), B(3, k), and C(-1, 4).
If the points are collinear, then Slope of AB = Slope of BC.
The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

Step 3: Detailed Explanation:
Calculate the slope of the line segment AB: \[ m_{AB} = \frac{k - 1}{3 - 1} = \frac{k - 1}{2} \] Calculate the slope of the line segment BC: \[ m_{BC} = \frac{4 - k}{-1 - 3} = \frac{4 - k}{-4} \] Set the slopes equal to each other: \[ \frac{k - 1}{2} = \frac{4 - k}{-4} \] Cross-multiply to solve for k: \[ -4(k - 1) = 2(4 - k) \] \[ -4k + 4 = 8 - 2k \] Move the terms with k to one side and the constant terms to the other: \[ 4 - 8 = -2k + 4k \] \[ -4 = 2k \] \[ k = \frac{-4}{2} = -2 \]

Step 4: Final Answer:
The value of k for which the points are collinear is -2.