Question

Let the harmonic mean of two positive real numbers $a$ and $b$ be $4$. If $q$ is a positive real number such that $a,5,q,b$ is an arithmetic progression, then the value(s) of $|q-a|$ is (are)

• A. $2$
• B. $3$
• C. $4$
• D. $5$

Answer 1

The correct option is A $2$

Given,

$a,5,q,b$ are in A.P.

$d=5-a\Rightarrow a=5-d$...(1)

$b=a+3d=(5-d)+3d\Rightarrow b=5+2d$...(2)

Given,

$4=\frac{2ab}{a+b}$

$2=\frac{ab}{a+b}$

From (1) and (2)

$2=\frac{(5-d)(5+2d)}{5-d+5+2d}$

$2=\frac{(5-d)(5+2d)}{10+d}$

$20+2d=25+5d-2d^2$

$2d^2-3d-5=0$

$(d+1)(2d-5)=0$

$\therefore d=-1,\frac{5}{2}$

$q=a+2d\Rightarrow q-a=2d$

$\therefore |q-a|=2|d|$

$\Rightarrow |q-a|=2|-1|=2$