x^2-z^2 is same as -(z^2-x^2)=
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Answer:
x
2
+y
2
+z
2
=xy+yz+zx=a
2
......... (i)
3x−y+z=a
3
........... (ii)
⟹2(x
2
+y
2
+z
2
)=2a
2
....... (iii)
2(xy+yz+zx)=2a
2
........ (iv) From (i)
Subtracting (iii) and (iv):−
2(x
2
+y
2
+z
2
)−2xy−2yz−2zx=0
x
2
+y
2
−2xy+y
2
+z
2
−2yz+z
2
+x
2
−2zx=0
(x−y)
2
+(y−z)
2
+(z−x)
2
=0
This is possible iff,
x−y=0,y−z=0,z−x=0
i.e. x=y,y=z,z=x
i.e. x=y=z=k (let)
In equation (ii)
3k−k+k=a
3
⇒k=
3
a
Hence, x=y=z=
3
a
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