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If a , b , c , are in A P ,a^2,b^2,c^2 a...

If `a , b , c ,` are in `A P ,a^2,b^2,c^2` are in HP, then prove that either `a=b=c` or `a , b ,-c/2` from a GP (2003, 4M)

Text Solution

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Since, a,b,c are in an AP
`:. 2b = a +c`
and `a^(2),b^(2) , c^(2)` are in HP
`rArr b^(2) = (2a^(2) c^(2))/(a^(2) + c^(2)) rArr ((a +c)/(2))^(2) = (2a^(2)c^(2))/(a^(2) + c^(2))`
`rArr (a^(2) + c^(2)) (a^(2) + c^(2) + 2ac) = 8a^(2) c^(2)`
`rArr (a^(2) + c^(2)) + 2ac (a^(2) + c^(2)) = 8a^(2) c^(2)`
`rArr (a^(2) + c^(2)) + 2ac (a^(2) + c^(2)) + a^(2)c^(2) = 9a^(2)c^(2)`
`rArr (a^(2) + c^(2) + ac)^(2) = 9a^(2) c^(2)`
`rArr a^(2) + c^(2) + ac = 3ac`
`rArr a^(2) + b^(2) - 2ac = 0`
`rArr (a -c)^(2) = 0 rArr a = c`
and if `a = c rArr b = c or a^(2) + c^(2) + ac = -3ac`
`rArr a^(2) + c^(2) + 2ac = -2ac`
`rArr (a +c)^(2) = -2ac`
`rArr 4b^(2) = -2ac rArr b^(2) = -(ac)/(2)`
Hence, `a, b, -(c)/(2)` are in GP
`:.` Either `a = b = c or a, b, -(c)/(2)` are in GP
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