If a, b and c (a > 0, c > 0) are in GP, then consider the following in respect of the equation ax² + bx + c = 0: 1. The equation has imaginary roots. 2. The ratio of the roots of the equation is 1 : ω where ω is a cube root of unity. 3. The product of roots of the equation is (b²/a²). Which of the statements given above are correct?

Since a, b, c are in GP, b² = ac. Discriminant = b² – 4ac = ac – 4ac = –3ac < 0 (since a, c > 0), so roots are imaginary. Statement 1 correct. Product of roots = c/a. Since b² = ac, c = b²/a, so c/a = b²/a². Statement 3 correct. For the ratio: roots are (–b ± i√3·√(ac))/(2a) = (–b ± i√3·b)/(2a) (since √(ac)=b for positive case, actually b could be negative). Roots are b(–1 ± i√3)/(2a) which are b·ω and b·ω² up to sign, where ω is a cube root of unity. Ratio = ω/ω² or equivalently 1:ω. Statement 2 correct. All three statements correct.