Question

If a, b, c are in G.P., then the roots of the equation $a{x}^2+bx+c=0$ are in the ratio

• A. $\frac{1}{2}(-1+i\sqrt{3})$
• B. $\frac{1}{2}(1-i\sqrt{3})$
• C. $\frac{1}{2}(-1-i\sqrt{3})$
• D. $\frac{1}{2}(1-+i\sqrt{3})$

Answer 1

Answer: (A), (C)

$\frac{1}{2}(-1+i\sqrt{3})$
$\frac{1}{2}(-1-i\sqrt{3})$

Given, $a,b,c$ are in G.P.

$\therefore b^2=ac$

or ${\left(\frac{b}{a}\right)}^2=\frac{c}{a}$

or ${(-\frac{b}{a})}^2=\frac{c}{a}$

$\therefore (\alpha +\beta {)}^2=\alpha \beta$

or ${\alpha }^2+{\beta }^2+\alpha \beta =0$

or ${\left(\frac{\alpha }{\beta }\right)}^2+\left(\frac{\alpha }{\beta }\right)+1=0$

or $\frac{\alpha }{\beta }=\frac{-1\pm \sqrt{3}i}{2}$ (where $i=\sqrt{-1}$)

Hence, options 'A' and 'C' are correct.