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if a,b,c are 3 distinct terms in a.p and a^2,b^2,c^2 are in h.p then a:b:c

if a,b,c are 3 distinct terms in a.p and a^2,b^2,c^2 are in h.p then a:b:c

Grade:11

1 Answers

Vikas TU
14149 Points
7 years ago
2b = a + c
Dividing by a
2b/a = 1 + c/a ….....................(1)
 
and
 
2/b^2 = 1/a^2 + 1/c^2  
2(a^2 + c^2) = (b^2 + c^2) + (a^2 + b^2)
a^2 + c^2  =  2b^2 …...................(2)

dividing by a^2
1 + (c/a)^2 = 2(b/a)^2 …...............(3)

from en. (1)

1 + (2b/a – 1)^2 = 2(b/a)^2
1 + 4b^2/a^2 + 1 – 4b/a = 2b^2/a^2
2 + 2b^2/a^2 – 4b/a = 0
let b/a = t
then, 
2 + 2t^2 – 4t = 0
t^2 -2t + 1 = 0
t^2 -t -t +1 = 0
t(t – 1) – 1(t – 1) = 0
t = 1
b/a =1
or a/b = 1
or a:b =1 or a = b
in eqn. (A)

2b = a + c 
2b = b + c
b = c
.i.e.
a = b = c
1: 1 : 1 is the answer.
 

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