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^20 for all (x, y) R ^2-\ (0,0)\ \ . Then which of the following statements is (are) TRUE?

Question

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?

Quantrex Academy · Accepted Answer

Correct answer: D. For 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 a unique 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.

JEE Advanced Mathematics Question (2024) — Solution

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