Consider the two equations x^2 - 2ax + b^2 = | vedclass

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(D) For the first equation $x^2 - 2ax + b^2 = 0$,let the roots be $\alpha_1$ and $\beta_1$. The sum of the roots is $\alpha_1 + \beta_1 = 2a$. The ari

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None of these

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(D) For the first equation $x^2 - 2ax + b^2 = 0$,let the roots be $\alpha_1$ and $\beta_1$. The sum of the roots is $\alpha_1 + \beta_1 = 2a$. The arithmetic mean $(AM)$ of the roots is $\frac{\alpha_1 + \beta_1}{2} = \frac{2a}{2} = a$.For the second equation $x^2 - 2bx + a^2 = 0$,let the roots be $\alpha_2$ and $\beta_2$. The sum of the roots is $\alpha_2 + \beta_2 = 2b$. The arithmetic mean $(AM)$ of the roots is $\frac{\alpha_2 + \beta_2}{2} = \frac{2b}{2} = b$.The product of the roots of the second equation is $\alpha_2 \beta_2 = a^2$. The geometric mean $(GM)$ of the roots is $\sqrt{\alpha_2 \beta_2} = \sqrt{a^2} = |a|$.Comparing the results,the $AM$ of the roots of the first equation is $a$,which is not necessarily equal to the $AM$ of the roots of the second equation $(b)$ or the $GM$ of the roots of the second equation $(|a|)$. However,if we look at the options,the arithmetic mean of the first equation is $a$. The arithmetic mean of the second equation is $b$. Thus,the correct choice is $D$.

Consider the two equations $x^2 - 2ax + b^2 = 0$ and $x^2 - 2bx + a^2 = 0$. The arithmetic mean of the roots of the first equation is equal to what?

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