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Prove that A.M of the roots of ${x^2} - 2ax + {b^2} = 0$ is equal to the geometric mean of the roots of the equation ${x^2} - 2bx + {a^2} = 0$, and vice-versa.

Last updated date: 19th Mar 2023
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Hint: A.M here means arithmetic mean, For this question we have to use the properties of roots of quadratic equations. These properties include the sum of roots and product of roots, equate these properties to the equation given above and then proceed.

We know the standard quadratic equation $a{x^2} + bx + c = 0$ and let its roots be α and β.
Properties α + β = $\dfrac{{ - b}}{a}$……………..(1)
α×β = $\dfrac{c}{a}$………………….(2)
Taking the equation ${x^2} - 2ax + {b^2} = 0$, let its roots be m and n
Therefore using equation (1) we get
$\Rightarrow m + n = \dfrac{{ - ( - 2a)}}{1}$
$\Rightarrow m + n = 2a$………..(3)
Since we have to find the arithmetic mean of the equation ${x^2} - 2ax + {b^2} = 0$
∴A.M will be $\dfrac{{m + n}}{2}$ using (3) in it we get
∴$\dfrac{{2a}}{2}$= a
∴ A.M of ${x^2} - 2ax + {b^2} = 0$ is a.
Let the roots of ${x^2} - 2bx + {a^2} = 0$ be p and q
Now using the property of product of roots i.e. equation (2)
$\Rightarrow$p×q = $\dfrac{{{a^2}}}{1}$
$\Rightarrow \sqrt {p \times q} = a$
This shows that A.M of ${x^2} - 2ax + {b^2} = 0$= G.M of ${x^2} - 2bx + {a^2} = 0$

Note: We know that standard Quadratic equation is $a{x^2} + bx + c = 0$, where a is the coefficient of${x^2}$, b is the coefficient of x and c is the constant and a≠0, since, if a=0, then the equation will no longer remain a quadratic.