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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) |
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Answer» Given, `a,b,c` are in AP. `therefore " " b=(a+c)/(2) " " "……(i)` and `a^(2),b^(2),c^(2)` in HP. `therefore " " b^(2)=(2a^(2)c^(2))/(a^(2)+c^(2)) " " "………(ii)` From Eq. (ii) `b^(2){(a+c)^(2)-2ac}=2a^(2)c^(2)` ` implies b^(2){(2b)^(2)-2ac}=2a^(2)c^(2) " " [" from Eq. (i)"]` `implies 2b^(4)-acb^(2)-a^(2)c^(2)=0` `implies (2b^(2)+ac)(b^(2)-ac)=0` `implies 2b^(2)+ac =0` or `b^(2)-ac=0` If `2b^(2)+ac=0`, than `b^(2)=-(1)/(2)ac` or `-(a)/(2),b,c` are in GP and if `b^(2)-ac =0 implies a,b,c` are in GP. But given, `a,b,c` are in AP. Which is possible only when `a=b=c`. |
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