Question

The quadratic equation whose roots are the $AM$ and $HM$ 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. none of these

Answer 1

The correct option is C $14x^2+45x-14=0$

Let $\alpha ,\beta$ be the roots of equation $x^2+7x-1=0$.
Hence
$\alpha +\beta =-7$ and $\alpha \beta =-1$
Now
AM of the roots will be
$\frac{\alpha +\beta }{2}$
$=\frac{-7}{2}$ ...(i)
HM of the roots will be
$\frac{1}{HM}=\frac{1}{2}(\frac{1}{\alpha }+\frac{1}{\beta })$
$=\frac{\alpha +\beta }{2\alpha \beta }$
$=\frac{7}{2}$
Or
$HM=\frac{2}{7}$
Now
${\alpha }^{\prime }=\frac{-7}{2}$ and ${\beta }^{\prime }=\frac{2}{7}$
Hence the equation with above roots will be
$(x+\frac{7}{2})(x-\frac{2}{7})=0$
$x^2+x(\frac{7}{2}-\frac{2}{7})-1=0$
$x^2+x\left(\frac{49-4}{14}\right)-1=0$
$x^2+\frac{45x}{14}-1=0$
$14x^2+45x-14=0$.