Question

The $A.M$. of the roots of a quadratic equation is $A$ and $G.M$. of its roots is $G$. The quadratic equation is

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

Answer 1

Answer: (D)

Let the equation be
$x^{2} - \left( \alpha+\beta\right)x + \alpha\beta = 0$ has roots $\alpha, \beta $
Also, $A= \frac{\alpha+\beta}{2}, G = \sqrt{\alpha\beta} $
$\Rightarrow \alpha + \beta = 2 \,A,$ $\alpha\beta= G^{2}$
$\therefore $ The required equation is $x^{2} -2\,Ax + G^{2} = 0$.