Question

If three points (8, 1), (k,-4) and (2,-5) are collinear, then k=

• A. 4
• B. 3
• C. 2
• D. 1

Answer 1

The Correct Option is B

To solve the problem, we need to determine the value of \( k \) for which the three points \( A(8, 1) \), \( B(k, -4) \), and \( C(2, -5) \) are collinear.

1. Understanding the Condition for Collinearity:
Three points are collinear if the area of the triangle formed by them is zero. The area of triangle formed by three points \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) is given by:

\( \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \)

2. Substituting the Given Points:
Let \( A = (8, 1) \), \( B = (k, -4) \), and \( C = (2, -5) \).

Using the formula:

\( \text{Area} = \frac{1}{2} \left| 8(-4 + 5) + k(-5 - 1) + 2(1 + 4) \right| = 0 \)

3. Simplifying the Expression:
\( \frac{1}{2} \left| 8(1) + k(-6) + 2(5) \right| = 0 \)
\( \frac{1}{2} \left| 8 - 6k + 10 \right| = 0 \)
\( \frac{1}{2} \left| 18 - 6k \right| = 0 \)

4. Solving the Equation:
Since the absolute value is zero:
\( 18 - 6k = 0 \)
\( 6k = 18 \)
\( k = 3 \)

Final Answer:
The value of \( k \) is \({3} \).