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
$x^2-\left(A+H\right)x+AH=0\Rightarrow x^2-x\left(-\frac{7}{2}+\frac{2}{7}\right)+\left(-\frac{7}{2}\right)\frac{2}{7}=0$
$\Rightarrow x^2+\frac{45}{14}x-1=0\Rightarrow 14x^2+45x-14=0$