Explanation

(b)
Since, $m$ is a root of the given equation
$\therefore\ (1 - ab)m^{2} - \left( a^{2} + b^{2} \right)m - (1 + ab) = 0$
$\Rightarrow \ m\left( a^{2} + b^{2} \right) + \left( m^{2} + 1 \right)ab = m^{2} - 1$ …(i)
Now, $\ H_{1} =$ first HM between $a$ and $b$
$= \frac{(m + 1)ab}{a + mb}$
And $H_{m} = \frac{(m + 1)ab}{b + ma}$
$\therefore H_{m} - H_{1} = (m + 1)ab\left\lbrack \frac{1}{b + ma} - \frac{1}{a + mb} \right\rbrack$
$= (m + 1)ab\frac{\lbrack(m - 1)(b - a)\rbrack}{(b + ma)(a + mb)}$
$= \ \frac{\left( m^{2} - 1 \right)ab((b - a)}{m\left( a^{2}{+ b}^{2} \right) + \left( m^{2} + 1 \right)ab}$
$= \ \frac{\left( m^{2} - 1 \right)ab(b - a)}{m^{2} - 1}$ [from Eq.(i)]
$= ab(b - a)$