Question

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

• A. are real and negative
• B. have negative real parts
• C. are equal
• D. have negative imaginary parts

Answer 1

The correct option is B have negative real parts

Since a,b,c are in G.P, hence

$a=a_0$

$b=a_{0}r$
$c=a_0{r}^2$

Since $a,b,c$ are positive, hence $a_0>0$ and $r>0$.

Therefore

$a{x}^2+bx+c=0$ implies

$a_0{x}^2+a_{0}rx+a_0{r}^2=0$

$x^2+rx+r^2=0$

$D=B^2-4AC$

$=r^2-4r^2$

$=-3r^2$

Hence

$D<0$.

Thus the above equation has imaginary roots.

$x=\frac{-r\pm \sqrt{-3r^2}}{2}$

$x=\frac{-r\pm ir\sqrt{3}}{2}$.

Since $r>0$ hence both the roots have negative real parts.