Arithmetic Mean

Let the roots of the equation

x%5E2%20-%202ax%20%2B%20b%20%3D%200

r_1

and

r_2

. Using Vieta's formulas, we know that:

r_1%20%2B%20r_2%20%3D%202a%20%5Cquad%20%5Ctext%7Band%7D%20%5Cquad%20r_1r_2%20%3D%20b.

The A.M. (Arithmetic Mean) of the roots is:

A%20%3D%20%5Cfrac%7Br_1%20%2B%20r_2%7D%7B2%7D%20%3D%20%5Cfrac%7B2a%7D%7B2%7D%20%3D%20a.

Next, let the roots of the equation

x%5E2%20-%202bx%20%2B%20a%5E2%20%3D%200

s_1

and

s_2

. Using Vieta's formulas again:

s_1%20%2B%20s_2%20%3D%202b%20%5Cquad%20%5Ctext%7Band%7D%20%5Cquad%20s_1s_2%20%3D%20a%5E2.

The G.M. (Geometric Mean) of the roots is:

G%20%3D%20%5Csqrt%7Bs_1s_2%7D%20%3D%20%5Csqrt%7Ba%5E2%7D%20%3D%20a.

Therefore,

A%20%3D%20a

and

G%20%3D%20a

. Since both the A.M. and G.M. are equal to

a

, the correct answer is

A%20%3D%20G

Chapter Questions
1 MCQs