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JEE Main20241 Feb 2024Evening ShiftMathematicsStraight LinesActual

Let A B C be an isosceles triangle in which A is at − 1 , 0 , ∠ A = 2 π 3 , A B = A C and B is on the positive x - axis. If B C = 4 3 and the line B C intersects the line y = x + 3 at α , β , then β 4 α 2 is:

Correct answer

36

Step-by-step solution

Let, B ≡ p , 0 ⇒ A B = p + 1 , p > 1 Now, finding point C using parametric form we get, x + 1 cos 2 π 3 = y - 0 sin 2 π 3 = p + 1 ⇒ C ≡ - p 2 - 3 2 , 3 2 p + 1 ⇒ B C 2 = - 3 2 p + 1 2 + 3 2 p + 1 2 ⇒ B C = 3 p + 1 = 4 3 ⇒ p = 3 So, C ≡ - 3 , 2 3 Now, equation of line B C is given by, y = - 1 3 x - 3 Solving with line y = x + 3 . ⇒ x + 3 = 3 - x 3 ⇒ x 3 + 3 3 = 3 - x ⇒ x 3 + 1 = 3 1 - 3 ⇒ x = 3 1 - 3 1 + 3 ⇒ x = - 3 2 4 - 2 3 ⇒ α = - 3 2 - 3 and β = - 3 + 3 3 ⇒ β 4 α 2 = 81 3 - 1 4 9 2 - 3 2 ⇒ β 4 α 2 = 9 4 - 2 3 2

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