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If x,y,z are real numbers satisfying the...

If x,y,z are real numbers satisfying the equation `25(9x^2+y^2)+9z^2-15(5xy+yz+3zx)=0` then prove that x,y,z are in AP.

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We hare,
`25(9x^(2)+y^(2))+9z^(2)-15(5xy+yz+3zx)=0`
`rArr (15x)^(2)+(5y)^(2)+(3z)^(2)-(15x)(5y)-(5y)(3z)-(3z)(15x)=0`
`rArr (1)/(2)[2(15x)^(2)+2(5y)^(2)+2(3z)^(2)-2(15x)(5y)-2(5y)(3z)-2(3z)(15x)]=0`
`rArr (1)/(2)[(15x-5y)^(2)+(5y-3z)^(2)+(3z-15x)^(2)]=0`
`rArr (15x-5y)^(2)+(5y-3z)^(2)+(3z-15x)^(2)=0`
It is possible only when
`15x-5y=0 and 5y-3z=0 and 3z-15x=0`
`rArr 15x=5y=3z`
`rArr (x)/(1)=(y)/(3)=(z)/(5)=k` (say)
`:. x=k, y=3k, z=5k`
Now, `y-x=3k-k=2k`
and `z-y=5k-3k=2k`
Since, `y-x=z-y`
`:. x,y,z ` are in A.P. `" "` Hence Proved
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