1. Given data

• $A=(4,6)$
• $B=(\alpha,\beta)$
• Side $AC$ lies on the line $2x-y=2$
• The angle bisector at $B$ is $y=x$
• Also, $2AB=BC$

We need to find $\alpha+2\beta$.

2. Find point $C$ using the side $AC$

Since $A$ lies on side $AC$, check that $A=(4,6)$ satisfies $2x-y=2$: $2(4)-6=8-6=2$ So $A$ is indeed on the line $AC$.

Because $y=x$ is the bisector of angle $B$, the two sides through $B$, namely $BA$ and $BC$, must be symmetric about the line $y=x$.

So first reflect point $A=(4,6)$ across the line $y=x$. Reflection across $y=x$ swaps coordinates, hence the reflected point is $A'=(6,4)$ Therefore, line $BC$ is the line through $B$ and $A'$, while line $BA$ is the line through $B$ and $A$.

Now point $C$ lies on line $BC$ and also on line $AC: 2x-y=2$.

3. Use the ratio condition $2AB=BC$

Since $A,B,C$ are on the two rays forming angle $B$, and $BA$ and $BC$ are symmetric about $y=x$, we can use the reflection idea.

Let $C'$ be the reflection of $C$ across $y=x$. Then $C'$ lies on line $BA$, and because reflection preserves distance from $B$, $BC=BC'$ Given $BC=2AB$ So along the line $BA$, the point $C'$ must satisfy $BC'=2AB$ Since $A$ is on ray $BA$, this means $C'$ is the point on line $BA$ such that $\overrightarrow{BC'}=2\overrightarrow{BA}$ Hence $C'=B+2(A-B)=2A-B$ Now reflect back across $y=x$ to get $C$.

Let $B=(\alpha,\beta)$. Then $2A-B=(8-\alpha,12-\beta)$ Reflecting across $y=x$ gives $C=(12-\beta,\,8-\alpha)$

4. Use that $C$ lies on $AC: 2x-y=2$

Substitute $C=(12-\beta,8-\alpha)$ into $2x-y=2$: $2(12-\beta)-(8-\alpha)=2$ $24-2\beta-8+\alpha=2$ $16+\alpha-2\beta=2$ $\alpha-2\beta=-14 \qquad (1)$

5. Use that $B$ lies on the angle bisector condition

Since line $BA$ reflects to line $BC$ about $y=x$, the point $B$ must lie on the bisector itself. Hence $B$ lies on $y=x$, so $\alpha=\beta$ Substitute into (1): $\alpha-2\alpha=-14$ $-\alpha=-14$ $\alpha=14$ Thus $\beta=14$

So $B=(14,14)$

6. Compute the required value

$\alpha+2\beta=14+2(14)=42$

7. Check with options

The value is $42$ So the correct option is A.

8. Comparison with stored answer

Stored correct answer: A

Our derived answer also gives A.