Let $\mathbb{R}^2$ denote $\mathbb{R} imes \mathbb{R}$. Let $S=\left\{(a, b, c): a, b, c \in \mathbb{R} ext { and } a x^2+2 b x y+c y^20 ext { for all }(x, y) \in \mathbb{R}^2-\{(0,0)\} ight\} .$ Then which of the following statements is (are) TRUE?

Question

Let $\mathbb{R}^2$ denote $\mathbb{R} 	imes \mathbb{R}$. Let $S=\left\{(a, b, c): a, b, c \in \mathbb{R} 	ext { and } a x^2+2 b x y+c y^2>0 	ext { for all }(x, y) \in \mathbb{R}^2-\{(0,0)\}ight\} .$ Then which of the following statements is (are) TRUE?

Aicharya · Accepted Answer

$\begin{aligned} & a x^2+2 b x y+c y^2>0 \ & y, x \in \mathbb{R}-\{(0,0)\} \ & \Rightarrow c\left(\frac{y}{x}\right)^2+2 b\left(\frac{y}{x}\right)+a>0 \ & 4 b^2-4 a c 2 \times 6$ $\therefore$ option A is incorrect $\begin{aligned} & \text {(B) if }\left(3, b, \frac{1}{12}\right) \in S \ & \Rightarrow b^2 0 \end{aligned}$ $\therefore$ unique solution $\therefore$ option D is correct.

Let $\mathbb{R}^2$ denote $\mathbb{R} \times \mathbb{R}$. Let $S=\left\{(a, b, c): a, b, c \in \mathbb{R}…

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