Q.

\(\alpha\) and \(\beta\) are the roots of the equation \(^ + bx + c = 0\). What is the condition for which \(\beta = 5 \alpha\)?

A. \(^=24ac\)

B. \(^=12ac\)

C. \(^=16ac\)

D. \(^=36ac\)

Use properties of roots.

D. \(^=36ac\)

For roots we know, \(\alpha+\beta=\frac \) and \(\alpha\beta=\frac \)

and we know \(\beta =5\alpha\), substituting this in the above relations we get

\(\Rightarrow 6\alpha=\frac \)    [∵ \(5\alpha+\alpha=6\alpha\)]

\(36^=\frac{^}{^} ………\text\)

\(\Rightarrow\alpha\beta=\frac \)

\(^=\frac\)   [∵ \(\alpha\times5\beta=^\)]

\(^=\frac………\text\)

⇒ combining (i) and (ii) we get,

\(36\times\frac=\frac{^}{}\)

\(36ac=5^\)