Question

If the points \( A(5, k) \), \( B(-3, 1) \) and \( C(-7, -2) \) are collinear, then \( k = \)

• A. 7
• B. \( -\frac{1}{7} \)
• C. \( \frac{1}{7} \)
• D. -7

Hint:

To check for collinearity, use the determinant method to calculate the area formed by the three points. If the area is zero, the points are collinear.

Answer 1

The Correct Option is A

Step 1: Use the condition for collinearity.
For three points to be collinear, the area of the triangle formed by them should be zero. The area can be calculated using the determinant method: \[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| = 0 \] Substitute the coordinates of \( A \), \( B \), and \( C \) into this equation.
Step 2: Solve for \( k \).
After simplifying, we find that \( k = 7 \).
Step 3: Conclusion.
The correct answer is (A) 7.