Let A ( 1 , 0 ) , B ( 2 , - 1 ) and C ( 73 , 43 ) be three points. If the equation of the bisector of the angle A B C is x + y = 5 , then the value of 2 + 2 is [2026]

Question

Let $A (1, 0), B (2, - 1)$ and $C (\frac{7}{3}, \frac{4}{3})$ be three points. If the equation of the bisector of the angle $A B C$ is $α x + β y = 5,$ then the value of $α^{2} + β^{2}$ is [2026]

Smart Achievers · Accepted Answer

(1)

$\frac{B D}{D C} = \frac{A B}{A C} = \frac{\sqrt{2} \times 3}{5 \sqrt{2}} = \frac{3}{5}$

$D = (\frac{12}{8}, \frac{4}{8}) = (\frac{3}{2}, \frac{1}{2})$

$Slope of A D = \frac{- 3 / 2}{1 / 2} = - 3$

$3 x + y = 5$

$α = 3, β = 1; α^{2} + β^{2} = 10$

Solution

Q.
Let $A (1, 0), B (2, - 1)$ and $C (\frac{7}{3}, \frac{4}{3})$ be three points. If the equation of the bisector of the angle $A B C$ is $α x + β y = 5,$ then the value of $α^{2} + β^{2}$ is [2026]

1 10

2 5

3 8

4 13

Ans.
(1)

$\frac{B D}{D C} = \frac{A B}{A C} = \frac{\sqrt{2} \times 3}{5 \sqrt{2}} = \frac{3}{5}$

$D = (\frac{12}{8}, \frac{4}{8}) = (\frac{3}{2}, \frac{1}{2})$

$Slope of A D = \frac{- 3 / 2}{1 / 2} = - 3$

$3 x + y = 5$

$α = 3, β = 1; α^{2} + β^{2} = 10$

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