Question

if the roorts of the equation $a{x}^2-4x+a^2=0$ are imaginary and the sum of the roots is equal to their product hen $a$ is:

• A. $-2$
• B. $4$
• C. $2$
• D. $Noneofthese$

Answer 1

The correct option is C $2$

$a{x}^2-4x+a^2=0$

since roots are imaginary

$D<0$

$\Rightarrow {\left(-4\right)}^2-4\times a\times a^2<0$

$\Rightarrow 16-4a^3<0$

$\Rightarrow 4a^3>16$

$\Rightarrow a^3>4$

Also Sum of roots $=$ product of roots

$\Rightarrow \frac{-\left(-4\right)}{a}=\frac{a^2}{a}$

$\Rightarrow 4=a^2$

$\Rightarrow a^2=4$

$\Rightarrow a=\pm 2$

If $a=2a^3=8$

If $a=-2,a^3={\left(-2\right)}^3=-8$

If $a^3>4a=2$

Hence $a=2$