In a △ABC, suppose y = x is the equation of the bisector of the angle B and the equation of the side

Question

In a \(\triangle \mathrm{ABC}\), suppose \(y=x\) is the equation of the bisector of the angle \(\mathrm{B}\) and the equation of the side \(\mathrm{A C}\) is \(2 x-y=2\). If \(\mathrm{2 A B=B C}\) and the point \(\mathrm{A}\) and \(\mathrm{B}\) are respectively \((4,6)\) and \((\alpha, \beta)\), then \(\alpha+2 \beta\) is equal to

(1) 42

(2) 39

(3) 48

(4) 45

Answer 1

Correct option is (1) 42

\(\mathrm{AD}: \mathrm{DC}=1: 2\)

\(\frac{4-\alpha}{6-\alpha}=\frac{10}{8}\)

\(\alpha=\beta\)

\(\alpha=14\text{ and }\beta=14\)