`a ,b ,a n dc` are all different and non-zero real numbers on arithmetic progression. If the roots of quadratic equation `a x^2+b x+c=0a r ealphaa n dbeta` such that `1/alpha+1/beta,alpha+beta,a n dalpha^2+beta^2` are in geometric progression the value of a/c will be_____.

Answer

3

Explanation

We have `(alpha+beta)^(2)=((1)/(alpha)+(1)/(beta))(alpha^(2)+beta^(2))`
`implies(alpha+beta)^(2)=((1)/(alpha)+(1)/(beta))[(alpha+beta)^(2)-2alphabeta]`
Substiting `alpha+beta=-(b)/(a)andalphabeta=(c)/(a)` we have
`impliescb^(2)+b(b^(2)-2ac)=0`
`bne0,:.bc+b^(2)-2ac=0` a, b, c are in AP, `:.b=(a+c)/(2)`
Therefore, we have `((a+c)c)/(2)+((a+c)/(2))^(2)-2ac=0`
`impliesa^(2)-4ac+3c^(2)=0or(a-c)(a-3c)=0`
`anec:.a=3c:.(a)/(c)=3`