JEE Advanced 2024 Paper 1Question 1
Let $\mathbb{R}^2$ denote $\mathbb{R}\times\mathbb{R}$. Let $S=\{(a,b,c): a,b,c\in\mathbb{R}$ and $ax^2+2bxy+cy^2>0$ for all $(x,y)\in\mathbb{R}^2-\{(0,0)\}\}$. Then which of the following statements is (are) TRUE?
Determinant
Solution
The form is positive definite iff $a>0$ and $ac>b^2$. (A): $ac=12<\frac{49}4=b^2$, no. (B): $3\cdot\frac1{12}>b^2\Rightarrow|2b|<1$, yes. (C): the determinant $ac-b^2>0$, unique solution. (D): $(a+1)(c+1)-b^2=(ac-b^2)+a+c+1>0$, unique solution.