Question

Show that - (x-y)²+(y-z)²+(z-x)² = 2(x-y)(x-z)+2(y-z)(y-x)+2(z-x)(z-y)​

Answer 1

Answer:

it is given that x y z are the AP therefore

y−x=z−y=d(say)

Now, (x 2 +zx+z 2 )−(x 2 +xy+y 2 )

=(−y 2+z2+x(z−y)=(z−y(x+y+z)=d(x+y+z)

and, (z 2 +yz+y 2 )−(z 2 +zx+x 2 )=(y 2 −x 2 )+z(y−x)=(y−x)(x+y+z)=d(x+y+z)

∴(x 2 +zx+z 2 )−(x 2 +xy+y 2 )=(z 2+yz+y 2 )−(z 2 +zx+x 2)

⇒x 2 +xy+y 2 ,z 2zx+x 2 ,y 2 +yz+z 2are in A.P.