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Q.If positive numbers a, b, c are in A.P. and a^2, b^2, c^2 are in H.P., then (A) a = b = c (B) 2b = a + c (C) b^2=whole underroot of (ac/8) i got the option the (b) ,since its a multiple choice Q so...pls guide.

Q.If positive numbers a, b, c are in A.P. and a^2, b^2, c^2 are in H.P., then
(A) a = b = c (B) 2b = a + c (C) b^2=whole underroot of (ac/8)

i got the option the (b) ,since its a multiple choice Q so...pls guide.

Grade:11

1 Answers

Sunil Raikwar
askIITians Faculty 45 Points
7 years ago
a,b,c are in AP
therefore
2b=a+c
a^2, b^2, c^2 are in HP
b^2=2a^2c^2/a^2+c^2
(a+c)^2/4=2a^2c^2/a^2+c^2
(a^2+c^2+2ac)(a^2+c^2)=8a^2c^2
(a^2+c^2)^2+2a^3c+2ac^3=8a^2c^2
a^4+c^4-6a^2c^2+2a^3c+2ac^3=0
(a^2-c^2)^2+2ac(a^2+c^2-2ac)=0
(a-c)^2{(a+c)^2+2ac}=0
but a, b, c are positive integers therefore a=c
therefore a=b=c

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