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? [2024]

Question

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

Smart Achievers · Accepted Answer

(2, 3, 4)

Given that $a x^{2} + 2 b x y + c y^{2} > 0$

$and x, y \in ℝ - {0}$

$\Rightarrow c {(\frac{y}{x})}^{2} + 2 b (\frac{y}{x}) + a > 0 \Rightarrow c > 0, D < 0$

$4 b^{2} - 4 a c < 0 \Rightarrow b^{2} < a c$

$(1) (2, \frac{7}{2}, 6)$
${(\frac{7}{2})}^{2} > 2 \times 6$
$∴$ $Option (1) is incorrect$

$(2) If (3, b, \frac{1}{12}) \in S$
$\Rightarrow b^{2} < 3 \cdot \frac{1}{12} \Rightarrow b^{2} < \frac{1}{4} \Rightarrow 4 b^{2} < 1$
$\Rightarrow | 2 b | < 1 Option (2) is correct$

$(3) a x + b y = 1$

$b x + c y = - 1$

$D = | \begin{matrix} a & b \\ b & c \end{matrix} | = a c - b^{2} \neq 0$

$∴$ $unique solution, option (3) is correct$

$(4) (a + 1) x + b y = 0$

$b x + (c + 1) y = 0$

$D = | \begin{matrix} a + 1 & b \\ b & c + 1 \end{matrix} |$

$= (a + 1) (c + 1) - b^{2} = a c - b^{2} + a + c + 1$

$Since a c - b^{2} > 0 \Rightarrow b^{2} < a c \Rightarrow a c is positive$

$\Rightarrow a and c are positive, then (a c - b^{2}) + a + c + 1 > 0$

$∴ unique solution \Rightarrow option (4) is correct$

Solution

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

1 $(2, \frac{7}{2}, 6) \in S$

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

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

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

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Solution

Q. 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 [2024]

1 (2,72,6)∈S

2 If (3,b,112)∈S, then |2b|<1.

3 For any given (a,b,c)∈S, the system of linear equations ax+by=1, by+cy=-1 has a unique solution.