First slide

Harmonic Progression

Question

If $x^{2} + 9 y^{2} + 25 z^{2} = xyz (\frac{15}{x} + \frac{5}{y} + \frac{3}{z})$ , then

Moderate

A

x, y, and z are in H.P.
B

$\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$ are in A.P.
C

x, y, z are in G.P.
D

$\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$ are in G.P.

Solution

$x^{2} + 9 y^{2} + 25 z^{2} = 15 yz + 5 zx + 3 xy$
$\begin{array}{l} \Rightarrow (x)^{2} + (3 y)^{2} + (5 z)^{2} - (x) (3 y) - (3 y) (5 z) - (x) (5 z) = 0 \\ \Rightarrow \frac{1}{2} [(x - 3 y)^{2} + (3 y - 5 z)^{2} + (x - 5 z)^{2}] = 0 \\ \Rightarrow x - 3 y = 0, 3 y - 5 z = 0, x - 5 z = 0 \\ x = 3 y = 5 z \\ \Rightarrow x : y : z = \frac{1}{1} : \frac{1}{3} : \frac{1}{5} \end{array}$
Therefore, 1/x, 1/y, and 1/z are in A.P. and x, y, and z are in H.P.

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