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:

Step-by-step explanation:

L.H.S. = (x -y )² + (y-z)² + (z-x)²

= x² + y ² - 2xy + y²+ z² - 2yz + z² + x² - 2zx

= 2x² + 2y² + 2z² - 2xy - 2yz - 2zx

R.H.S. = 2 (x-y) (x -z ) + 2 (y - z) ( y - x) + 2 (z -x) ( z - y)

= 2 ( x² - x z - xy + yz ) + 2 (y² - xy - yz +xz ) + 2 ( z² - yz -xz + xy )

= 2x² - 2xz - 2xy + 2yz + 2y² - 2xy - 2yz + 2xz + 2z² - 2yz - 2xz + 2xy

= 2x² + 2y² + 2z² - 4xy + 2xy - 4yz + 2yz - 4xz + 2xz

= 2x² + 2y² + 2z² - 2xy - 2yz - 2xz

∴ L.H.S. = R.H.S. is proved.