Question

The quadratic equation whose roots are the arithmetic mean and the harmonic mean of the roots of the equation $x^{2}+7x-1=0$ is

• A. $14x^{2}+14x-45=0$
• B. $45x^{2}-14x+14=0$
• C. $14x^{2}+45x-14=0$
• D. $45x^{2}+14x-45=0$

Answer 1

Answer: (C)

Let, $\alpha ,\beta $ are the roots of the equation $x^{2}+7x-1=0$
$\Rightarrow \alpha +\beta =-7,\alpha \beta =-1$
$\Rightarrow A=\frac{\alpha + \beta }{2}=-\frac{7}{2}$
and $H=\frac{2 \alpha \beta }{\alpha + \beta }=\frac{2 \left(- 1\right)}{\left(- 7\right)}=\frac{2}{7}.$
Now, equation whose roots are $A$ and $H$ is
$x^{2}-\left(A + H\right)x+AH=0\Rightarrow x^{2}-x\left(- \frac{7}{2} + \frac{2}{7}\right)+\left(- \frac{7}{2}\right)\frac{2}{7}=0$
$\Rightarrow x^{2}+\frac{45}{14}x-1=0\Rightarrow 14x^{2}+45x-14=0$