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JEE Advanced2024MathematicsStraight LinesActual

Let R ^2 denote R R . Let S= (a, b, c): a, b, c R and a x^2+2 b x y+c y^2>0 for all (x, y) R ^2- (0,0) . Then which of the following statements is (are) TRUE?

Options

  1. A(2, 7 2 , 6 ) S
  2. BIf (3, b, 1 12 ) S , then |2 b| < 1 .
  3. CFor any given (a, b, c) S , the system of linear equations aligned & a x+b y=1 & b x+c y=-1 aligned has a uniq
  4. DFor any given (a, b, c) S , the system of linear equations aligned & (a+1) x+b y=0 & b x+(c+1) y=0 aligned has

Correct answer

B. If (3, b, 1 12 ) S , then |2 b| < 1 .

Step-by-step solution

( aligned & a x^2+2 b x y+c y^2>0 & y, x R - (0,0) & c ( y x )^2+2 b ( y x )+a>0 & 4 b^2-4 a c 2 6 ) ( ) option A is incorrect ( aligned & (B) if (3, b, 1 12 ) S & b^2 0 aligned ) ( ) unique solution ( ) option D is correct.

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