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(B) Given that $H$ is the harmonic mean between $a$ and $b$,we have $H = \frac{2ab}{a + b}$.Consider the expression $E = \frac{H + a}{H - a} + \frac{H

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$2$

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(B) Given that $H$ is the harmonic mean between $a$ and $b$,we have $H = \frac{2ab}{a + b}$.Consider the expression $E = \frac{H + a}{H - a} + \frac{H + b}{H - b}$.Substituting $H = \frac{2ab}{a + b}$ into the first term:$\frac{H + a}{H - a} = \frac{\frac{2ab}{a + b} + a}{\frac{2ab}{a + b} - a} = \frac{2ab + a(a + b)}{2ab - a(a + b)} = \frac{2ab + a^2 + ab}{2ab - a^2 - ab} = \frac{a^2 + 3ab}{ab - a^2} = \frac{a(a + 3b)}{a(b - a)} = \frac{a + 3b}{b - a}$.Similarly,for the second term:$\frac{H + b}{H - b} = \frac{\frac{2ab}{a + b} + b}{\frac{2ab}{a + b} - b} = \frac{2ab + b(a + b)}{2ab - b(a + b)} = \frac{2ab + ab + b^2}{2ab - ab - b^2} = \frac{3ab + b^2}{ab - b^2} = \frac{b(3a + b)}{b(a - b)} = \frac{3a + b}{a - b} = -\frac{3a + b}{b - a}$.Adding the two terms:$E = \frac{a + 3b}{b - a} - \frac{3a + b}{b - a} = \frac{a + 3b - 3a - b}{b - a} = \frac{2b - 2a}{b - a} = \frac{2(b - a)}{b - a} = 2$.

If the harmonic mean between $a$ and $b$ is $H$,then $\frac{H + a}{H - a} + \frac{H + b}{H - b} = $

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