Question

In a △ABC, suppose y = x is the equation of the bisector of the angle B and the equation of the side AC is 2x−y = 2. If 2AB = BC and the points A and B are respectively (4, 6) and (α, β), then α + 2β is equal to:

• A. 42
• B. 39
• C. 48
• D. 45

Answer 1

The Correct Option is A

\(\text{Define Given Points and Conditions: Let } A = (4, 6), B = (\alpha, \beta), \text{ and } C = (-2, -6).\)
The angle bisector \( y = x \) passes through point \( D \), which divides \( AC \) in the ratio \( AD : DC = 1 : 2 \).

Set up the Ratio Condition: Since \( AD : DC = 1 : 2 \), the coordinates of \( D \) (which lies on \( y = x \)) can be calculated using the section formula:
\(D = \left( \frac{2 \cdot 4 + (-2)}{1 + 2}, \frac{2 \cdot 6 + (-6)}{1 + 2} \right) = (2, 2)\)
Equation of Side \( AC \): Since \( D \) lies on the bisector and divides \( AC \) such that \( 2AB = BC \), we set up equations using the distances:
\(\frac{4 - \alpha}{6 - \alpha} = \frac{10}{8}\)

Solve for \( \alpha \) and \( \beta \): Solving these equations gives:
\(\alpha = \beta = 14\)
Calculate \( \alpha + 2\beta \):
\(\alpha + 2\beta = 14 + 2 \times 14 = 42\)
So, the correct option is: \( 42 \)