If `alpha,beta` are the nonzero roots of `a x^2+b x+c=0a n dalpha^2,beta^2` are the roots of `a^2x^2+b^2x^2+b^2x+c^2=0,t h e na ,b ,c` are in a. G.P. b. H.P. c. A.P. d. none of these

Options

A. G.P.
B. H.P.
C. A.P.
D. none of these

Answer

1

Explanation

`alpha, beta` are roots of `ax^(2) + bx + c = 0`. Hence,
`alpha + beta = -b/a " "(1)`
`alphabeta = c/a" "(2)`
`alpha^(2), beta^(2)` are roots of `a^(2)x^(2) + b^(2)x + c^(2) = 0`. Hence,
`alpha^(2) + beta^(2) = -b^(2)/a^(2)" "(3)`
`alpha^(2)beta^(2) = c^(2)/a^(2)" "(4)`
Now, from (3), we have
`(alpha + beta)^(2) - 2alphabeta = - b^(2)/a^(2)`
`rArr ((-b)/(a))^(2) - 2c/a = (-b^(2))/(a^(2))`
or `2b^(2)/a^(2) = (2c)/(a)`
or `b^(2) = ac rArr a, b, c` are in G.P.