Question

If m is a root of the given equation $(1-ab)x^2-(a^2+b^2)x-(1+ab)=0$ and m harmonic means are inserted between a and b, then the difference between the last and the first of the means equals

• A. b - a
• B. ab(b - a)
• C. a(b - a)
• D. ab(a - b)

Answer 1

The correct option is B ab(b - a)

By the given condition
$(1-ab)m^2-(a^2+b^2)m-(1+ab)=0$
$\Rightarrow m(a^2+b^2)+(m^2+1)ab=m^2-1$ ......(i)
Now $H_1$ First H.M. between a and b

= $\frac{(m+1)ab}{a+mb}$ and $H_m=\frac{(m+1)ab}{b+ma}$
$\therefore H_m-H_1=(m+1)ab[\frac{1}{b+ma}-\frac{1}{a+mb}]$
$=(m+1)ab\frac{\left[\right(m-1\left)\right(b-a\left)\right]}{(b+ma)(a+mb)}=\frac{(m^2-1)ab(b-a)}{M(a^2+b^2)+(m^2+1)ab}$
$=\frac{(m^2-1)ab(b-a)}{m^2-1}\left[by\right(i\left)\right]$
= ab(b - a).