Progressions in an Tri-variable quadratic equation

Algebra Level 4

If

4 x 2 + 9 y 2 + 16 z 2 − 6 x y − 12 y z − 8 z x = 0 \large 4x^2 + 9y^2 + 16z^2 - 6xy - 12yz - 8zx = 0

then,

x , y , z x, y, z are in:

None of the these Arithmetic-Geometric Progression Geometric Progression Arithmetic Progression Harmonic Progression

Vishwak Srinivasan by Vishwak Srinivasan , 24, India

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1 solution

Spandan Senapati
Mar 17, 2017

Using the trivial inequality a 2 + b 2 + c 2 > = a b + b c + c a a^2+b^2+c^2>=ab+bc+ca And setting a = 2 x , b = 3 y , c = 4 z a=2x,b=3y,c=4z .Yields the equality condition so 2 x = 3 y = 4 z = k 2x=3y=4z=k .So x = k / 2 , y = k / 3 , z = k / 4 x=k/2,y=k/3,z=k/4 .And now clearly the reciprocals of x , y , z x,y,z being an A P AP the no x , y , z x,y,z form a H P HP .

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