Relation between A.M., G.M. and H.M. MCQ - Practice Questions & Answers

Concepts Covered - 1

Properties of A.M. and G.M.

Properties of A.M. and G.M

A and G are arithmetic and geometric mean of ‘a’ and ‘b’, two real, positive and distinct number. Then,

a and b are the roots of the equation x²-2Ax+G²=0.
a and b are given by $A \pm \sqrt{(A+G)(A-G)}$ .

Proof:

$\begin{array}{l}{A=\frac{a+b}{2} \Rightarrow 2 A=a+b} \\ {G=\sqrt{a b} \Rightarrow G^{2}=a b}\end{array}$

a and b are the roots of the equation, then

$\begin{array}{l}{x^{2}-2(\text { sum of roots) } x+\text { products of roots }=0} \\ {\Rightarrow x^{2}-(a+b)+a b=0} \\ {\Rightarrow x^{2}-2 A x+G^{2}=0}\end{array}$

Roots of the equation are

$\begin{array}{l}{x=\frac{2 A \pm \sqrt{(-2 A)^{2}-4 \cdot 1 \cdot G^{2}}}{2}} \\ {{x}=A \pm \sqrt{(A+G)(A-G)}}\end{array}$

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Relation between A.M., G.M. and H.M. - Practice Questions & MCQ

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