In a , 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 , then is equal to [2024]

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Solution

Q.
In a $△ A B C$ , 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 $α + 2 β$ is equal to [2024]

1 39

2 42

3 48

4 45

Ans.
(2)

We have, y = x ... (i)

and 2x – y = 2 ... (ii)

Solving (i) and (ii), we get x = 2 and y = 2

As, $△ A B D \tilde{=} △ C B D$

$∴ \frac{A B}{B C} = \frac{A D}{D C} = \frac{1}{2}$

$D (\frac{λ + 8}{3}, \frac{2 λ - 2 + 12}{3}) lies on y = x$

$∴ \frac{λ + 8}{3} = \frac{2 λ + 10}{3} \Rightarrow λ = - 2 . S o, C \equiv (- 2, - 6)$

The image of the point A will lies on the line BC.

Let A' = (6, 4)

Now, $\frac{β - 4}{α - 6} = \frac{10}{8} \Rightarrow \frac{α - 4}{α - 6} = \frac{5}{4} (∵ α = β)$

$\Rightarrow 4 α - 16 = 5 α - 30 \Rightarrow 14 = α = β$

$∴ α + 2 β = 14 + 2 \times 14 = 42$

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