Let \mathbb{R}^2 denote \mathbb{R} \times \mathbb{R}. Let S= \{(a, b, c): a, b, c \in \mathbb{R}. and a x^2+2 b x y+c y^2>0 for all (x, y) \in \mathbb{R}^2-\{(0,0)\}\}. Then which of the following statements is (are) TRUE? [JEE Adv 2024 P1]

If (3, b, \frac{1}{12}) \in S, then |2 b|< 1

For any given (a, b, c) \in S, the system of linear equations \table a x+b y=1; b x+c y=-1 has a unique solution.

For any given (a, b, c) \in S, the system of linear equations \table (a+1) x+b y=0 ; b x+(c+1) y=0 has a unique solution.

Matrices and Determinants

Section: Mathematics

Question : 4 of 56

Marks: +1, -0

Let ℝ2 denote ℝ×ℝ. Let
S={(a,b,c):a,b,c∈ℝ. and ax2+2bxy+cy2>0 for all (x,y)∈ℝ2−{(0,0)}}.
Then which of the following statements is (are) TRUE?

[JEE Adv 2024 P1]

(2,‌
7
2
,6)∈S
If (3,b,‌
1
12
)∈S, then |2b|<1
For any given (a,b,c)∈S, the system of linear equations
‌ax+by=1
‌bx+cy=−1
has a unique solution.
For any given (a,b,c)∈S, the system of linear equations
‌(a+1)x+by=0
‌bx+(c+1)y=0
has a unique solution.

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