Determinants · Question 1

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?

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 bx+cy=-1 has a unique solution.
4. 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.

Answer

(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 bx+cy=-1 has a unique solution., (4) 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.

Solution

ax2+2bxy+cy2>0 for all (x,y) (0,0) a>0 and ac-b2>0 (discriminant 4b2-4ac<0).

(A) ac=12<b2= 494. False.

(B) 3 112= 14>b2 |b|< 12 |2b|<1. True.

(C) Coefficient determinant ac-b2>0 0. Unique solution. True.

(D) Determinant (a+1)(c+1)-b2=(ac-b2)+a+c+1>0 (as a,c>0). Unique (trivial) solution. True.

Answer: (B), (C), (D)