KVPY Sample Paper KVPY Stream-SX Model Paper-15

If \[x,y,z\in {{R}^{+}}\] such that: \[{{x}^{2}}+9{{y}^{2}}+25{{z}^{2}}=xyz\left( \frac{15}{x}+\frac{5}{y}+\frac{3}{z} \right).\] Then x, y, z are in-

A) A.P.

B) G.P.

C) H.P.

D) None of these

Correct Answer: C

Solution :

We have, \[\frac{{{x}^{2}}+9{{y}^{2}}}{2}\ge 3xy,\]

\[\frac{{{x}^{2}}+25{{z}^{2}}}{2}\ge 5xz\] \[,\frac{9{{y}^{2}}+25{{z}^{2}}}{2}\ge 15xy\]

\[\Rightarrow {{x}^{2}}+9{{y}^{2}}+25{{z}^{2}}\ge xyz\left( \frac{15}{x}+\frac{5}{y}+\frac{3}{z} \right)\]

Here, the equality sign holds if and only if; \[x=3y=5z\]

\[\Rightarrow x,y,z\] are in H.P.