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JEE Main202429 Jan 2024Morning ShiftMathematicsStraight LinesActual

In a ∆ ABC , suppose y = x is the equation of the bisector of the angle B and the equation of the side A C is 2 x - y = 2 . If 2 A B = B C and the point A and B are respectively ( 4 , 6 ) and ( α , β ) , then α + 2 β is equal to

Options

  1. A- 4
  2. B42
  3. C2
  4. D- 1

Correct answer

B. 42

Step-by-step solution

Given: Line A C : 2 x - y = 2 and B D : x = y . Line A C and B D intersects at point D . ⇒ 2 x - x = 2 ⇒ x = y = 2 ⇒ D ≡ 2 , 2 We know that, by angle bisector theorem, the angle bisector of a triangle divides the base in the same ratio as the other two sides. ⇒ C D A D = B C A B It is given that, 2 A B = B C . ⇒ C D A D = 2 A B A B ⇒ C D : A D = 2 : 1 Let the coordinates of C be h , k . By using section formula, D ≡ h × 1 + 2 × 4 3 , 1 × k + 2 × 6 3 = 2 , 2 ⇒ h + 8 = 6 , k + 12 = 6 ⇒ h = - 2 , k = - 6 ⇒ C ≡ - 2 , -

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