If a b ,b c ,c a are in H.P., then - Vidyadip

MCQ

If $\frac{a}{b},\frac{b}{c},\frac{c}{a}$ are in $H.P.$, then

✓
${a^2}b,\,{c^2}a,\,{b^2}c$ are in $A.P.$
B
${a^2}b,\,{b^2}c,\,{c^2}a$ are in $H.P.$
C
${a^2}b,\,{b^2}c,\,{c^2}a$ are in $G.P.$
D
None of these

✓

Answer

Correct option: A.

${a^2}b,\,{c^2}a,\,{b^2}c$ are in $A.P.$

a
(a) $\frac{b}{a},\frac{c}{b},\frac{a}{c}$ are in $A.P.$

==> $\frac{{2c}}{b} = \frac{b}{a} + \frac{a}{c}$

$ \Rightarrow \frac{{2c}}{b} = \frac{{bc + {a^2}}}{{ac}}$

==> $2a{c^2} = {b^2}c + b{a^2}$

$\therefore \,{a^2}b,\,{c^2}a$ and ${b^2}c$ are in $A.P.$

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