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If m is the root of the (1-ab)x²-(a²+b²)x-(1+ab)=0 and if m harmonic means are inserted between A and B then the difference between the last and first of the means is equal to A )ab(a-b) B)a(b-a) C)ab(b+a) D)b-a

If m is the root of the (1-ab)x²-(a²+b²)x-(1+ab)=0 and if m harmonic means are inserted between A and B then the difference between the last and first of the means is equal to
A )ab(a-b)
B)a(b-a)
C)ab(b+a)
D)b-a
 

Grade:11

1 Answers

Arun
25750 Points
3 years ago
Let H 
1
​ 
 ,H 
2
​ 
 ,...,H 
9
​ 
  be the harmonic means between 2 and 3, then, H 
1
​ 
 ,H−2,...,H 
9
​ 
 ,3 are in H.P.
∴ 
2
1
​ 
 , 
H 
1
​ 
 
1
​ 
 , 
H 
2
​ 
 
1
​ 
 ,..., 
H 
9
​ 
 
1
​ 
 , 
3
1
​ 
  are in A.P. with common difference
D= 
(9+1)×2×3
2−3
​ 
 =− 
60
1
​ 
 [∵D= 
(n+1)ab
a−b
​ 
 ]
 
∴ 
H 
i
​ 
 
1
​ 
 = 
2
1
​ 
 +iD;i=1,2,3,...,9⇒ 
H 
i
​ 
 
1
​ 
 = 
2
1
​ 
 − 
60
i
​ 
 ⇒ 
H 
i
​ 
 
6
​ 
 =3− 
10
i
​ 
    ...(1)
Let A 
1
​ 
 ,A 
2
​ 
 ,...,A 
9
​ 
  be the nine A.M. 
′
 s between 2 and 3.
Then, 2,A 
1
​ 
 ,A 
2
​ 
 ,...,A 
9
​ 
 ,3 are in A.P. with common difference
d= 
9+1
3−2
​ 
 = 
10
1
​ 
 [∵d= 
n+1
b−a
​ 
 ]
 
∴A 
i
​ 
 =2+id;i=1,2,...,9⇒A 
i
​ 
 =2+ 
10
i
​ 
    ...(2)
 
From (1) and (2), we have
A 
i
​ 
 + 
H 
i
​ 
 
6
​ 
 =2+ 
10
i
​ 
 +3− 
10
i
​ 
 =5 for i=1,2,3,...,9

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