First slide

Harmonic Progression

Question

If X,Y, Z are real and $4 x^{2} + 9 y^{2} + 16 z^{2} - 6 xy - 12 yz - 8 zx = 0$ then x, y, z are in

Moderate

A

A.P.
B

G.P.
C

H.P
D

none of these

Solution

Multiplying the given expression by 2 and rewriting it, we have
$\begin{array}{l} (2 x - 3 y)^{2} + (3 y - 4 z)^{2} + (4 z - 2 x)^{2} = 0 \\ \Rightarrow 2 x = 3 y = 4 z \\ \Rightarrow \frac{1}{x}, \frac{1}{y}, \frac{1}{z} are in A.P. \\ \Rightarrow x, y, z are in H.P. \end{array}$

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