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. 3,4
B. 2,5
C. 3,6
D. 6,9

Answer 1

Correct Answer - B
It is given that the harmonic mean of a and b is 4.
`:." "(2ab)/(a+b)=4rArr" "ab=2(a+b)` . . .(i)
It is given that a,5,q,b are in A.P. with common difference d(say). Then,
5=a+d and b=a+3d
`rArr" "` a=5-d and b=5-d+3d
`rArr" "` a=5-d and b=5-d+3d
`rArr" "`a=5-d and b=5+2d
Substituting the values of a and b in (i), we obtain
(5-d)(5+2d)=2(5-d+5+2d)
`rArr" "25+5d-2d^(2)=20+2d`
`rArr" "2d^(2)-3d-5=0rArr(2d-5)(d+1)=0rArrd=-1,5//2`
`:." "q=a+2drArr|q-a|=|2d|=2,5`