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Question

# If x,y,z are real and 9x2+16y2+25z2−12xy−20yz−15zx=0, then 1x,1y,1z are in

A
arithmetic progression
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B
geometric progression
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C
harmonic progression
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D
None of the above
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Solution

## The correct option is A arithmetic progressionGiven : 9x2+16y2+25z2−12xy−20yz−15zx=0 ⇒(3x)2+(4y)2+(5z)2−(3x)(4y)−(4y)(5z)−(5z)(3x)=0 ⋯(i) It is of the form a2+b2+c2−ab−bc−ac=0 which can be written as 12[(a−b)2+(b−c)2+(c−a)2]=0 ⇒a=b=c Similarly, equation (i) can be written as (3x−4y)2+(4y−5z)2+(5z−3x)2=0 ⇒3x=4y=5z Let 3x=4y=5z=k ⇒x=k3,y=k4,z=k5 ∴1x,1y,1z are in A.P.

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