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JEE Main202427 Jan 2024Evening ShiftMathematicsStraight LinesActual

Let R be the interior region between the lines 3 x - y + 1 = 0 and x + 2 y - 5 = 0 containing the origin. The set of all values of a , for which the points a 2 , a + 1 lie in R , is :

Options

  1. A( - 3 , - 1 ) ∪ - 1 3 , 1
  2. B( - 3 , 0 ) ∪ 1 3 , 1
  3. C( - 3 , 0 ) ∪ 2 3 , 1
  4. D( - 3 , - 1 ) ∪ 1 3 , 1

Correct answer

B. ( - 3 , 0 ) ∪ 1 3 , 1

Step-by-step solution

Let, P ≡ a 2 , a + 1 , L 1 : 3 x - y + 1 = 0 and L 2 : x + 2 y - 5 = 0 . Now, origin and P lies on the same side with respect to line L 1 . ⇒ L 1 ( 0 ) · L 1 ( P ) > 0 ⇒ 3 a 2 - ( a + 1 ) + 1 > 0 ⇒ 3 a 2 - a > 0 ⇒ a 3 a - 1 > 0 ⇒ a ∈ ( - ∞ , 0 ) ∪ 1 3 , ∞ . . . i Now, L 2 : x + 2 y - 5 = 0 Origin and P lies on same side with respect to L 2 ⇒ L 2 ( 0 ) · L 2 ( P ) > 0 ⇒ a 2 + 2 ( a + 1 ) - 5 < 0 ⇒ a 2 + 2 a - 3 < 0 ⇒ ( a + 3 ) ( a - 1 ) < 0 ∴ a ∈ ( - 3 , 1 ) . . . ii Using intersection of i and i i , ⇒ a ∈ ( - 3 , 0

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