If \(a, b, c\) are in Arithmetic Progression (AP), then the roots of the equation \(a x^2-2 b x+c=0\) are
If \(a, b, c\) are in Arithmetic Progression (AP), then the roots of the equation \(a x^2-2 b x+c=0\) are
\(1, \frac{c}{a}\)
\(\frac{-1}{a},-c\)
\(-1, \frac{-c}{a}\)
\(-2, \frac{-c}{2 a}\)
Solution
Given, \(a, b, c\) are in AP
\(\begin{aligned}
2 b & =a+c \\
a x^2-2 b x+c & =0 \\
a x^2-(a+c) x+c & =0 \\
a x^2-a x-c x+c & =0 \\
a x(x-1)-c(x-1) & =0 \\
(x-1)(a x-c) & =0 \\
x & =1, \frac{\mathrm{c}}{\mathrm{a}}
\end{aligned}\)
Hence, option (a) is correct.