Question

If the ratio of the ratio of the roots of lx² + µx + v = 0 is equal to the ratio of the roots of x² + x + 1 = 0 then l, µ, v are in –

• A. A.P.
• B. G.P.
• C. H.P.
• D. None of these

Answer 1

Answer: (B)

G.P.

Answer 2

As ratio of roots for lx² + µx + v = 0 and x² + x + 1 = 0 are equal

\$\frac{\alpha}{\beta}$= $\frac{\alpha'}{\beta'}$, where a, b are roots of lx² + mx + v = 0 and a¢, b¢ are roots of x² + x + 1 = 0 gives by w and w²

\ $\frac{\alpha}{\beta}$= $\frac{\omega}{\omega^{2}}$= $\frac{1}{\omega}$ Ž b = wa

And a¢, b¢ are roots of x² + x + 1 = 0, given by w and w²

\$\frac{\alpha}{\beta}$= $\frac{\omega}{\omega^{2}}$ = $\frac{1}{\omega}$ Ž = wa

and a + b = – $\frac{µ}{\lambda}$, ab = $\frac{v}{\lambda}$

a (1 + w) = – $\frac{\mu}{\lambda}$, $\alpha^{2}$w = $\frac{v}{\lambda}$

Ž –aw² = $\frac{–µ}{\lambda}$, a²w = $\frac{v}{\lambda}$

Ž $\frac{µ^{2}}{\lambda^{2}}$ = $\frac{v}{\lambda}$ or µ² = lv.

Hence (2) is correct answer.