In the quadratic equation `ax^2 + bx + c = 0`. if `delta = b^2-4ac` and `alpha+beta , alpha^2+beta^2 , alpha^3+beta^3` are in G.P. and `alpha,beta` are the roots of `ax^2 + bx + c =0`

Options

A. `Delta!=0`
B. `c=0`
C. `cb!=0`
D. `Delta=0`

Answer

D

Explanation

`(alpha^(2)+beta^(2))=(alpha+beta)(alpha^(3)+beta^(3))`
`implies{(alpha+beta)^(2)-2alpha beta}^(2)=(alpha+beta){(alpha+beta)^(2)-2alpha beta(alpha +beta)}`
`=((b^(2))/(a^(2))-(2c)/a)^(2)=(-b/a)((-b^(3))/(a^(3))+(3bc)/(a^(2)))`
`implies((b^(2)-2ac)/(a^(2)))^(2)=((-b)/a)((-b^(3)+3abc)/(a^(3)))`
`implies4a^(2)c^(2)=acb^(2)`
`impliesac(b^(2)-4ac)=0`
As `a!=0`
`impliesc Delta=0`