Question

If \( C \left( \frac{\alpha}{8}, 4 \right) \) is the mid-point of the line joining the points \( A(-4, 2) \) and \( B(5, 6) \), then the value of \( \alpha \) will be:

• A. -8
• B. 4
• C. -4
• D. 2

Hint:

To find the midpoint of two points, use the formula \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).

Answer 1

The Correct Option is B

We are given that point \( C \left( \frac{\alpha}{8}, 4 \right) \) is the mid-point of the line joining points \( A(-4, 2) \) and \( B(5, 6) \). To find the value of \( \alpha \), we use the midpoint formula. The midpoint formula is: \[ C = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] where \( A(x_1, y_1) \) and \( B(x_2, y_2) \) are the coordinates of the two points.
Step 1: Apply the midpoint formula.
The coordinates of \( A \) are \( (-4, 2) \), and the coordinates of \( B \) are \( (5, 6) \). The midpoint \( C \) has coordinates \( \left( \frac{\alpha}{8}, 4 \right) \). Using the midpoint formula for the \( x \)-coordinate: \[ \frac{x_1 + x_2}{2} = \frac{-4 + 5}{2} = \frac{1}{2} \] Thus, we equate: \[ \frac{\alpha}{8} = \frac{1}{2} \]
Step 2: Solve for \( \alpha \).
Multiplying both sides by 8: \[ \alpha = 4 \]
Step 3: Conclusion.
Therefore, the value of \( \alpha \) is 4. The correct answer is (B).