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

Step-by-step solution

Given that a, b, c are in GP b^2=a c Now, a x^2+b x+c=0 aligned D & =b^2-4 a c & =a c-4 a c=-3 a c 0 aligned 1. Roots are imaginary (correct) Let, a=2, b=4, c=8 aligned x^2+2 x+4 & =0 x & = -2 4-16 2 =-1 3 aligned 2. Ratio of roots = -1+ 3 -1- 3 = ^2 = 1 (correct) 3. aligned Product of roots & = (-1+ 3 )(-1- 3 ) 2 4 & = ^2 4=4= b^2 a^2 aligned (correct) Hence, statements 1,2 and 3 are correct.