Question

5. If x + y + z = 6xyz and xyz = x + y + z, then
the value of (x - y) square + (x - 2)square + (z - x)square is equal
to
(1)
(3)
o
6
(2) 3
(4) 12
​

Answer 1

Answer :

x³ + y³ + z³ = 6xyz .... (i)

x + y + z = xyz ...... (ii)

We know that ;

x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)

Putting the value of (i),

⇒ 6xyz - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)

⇒ (x + y + z)(x² + y² + z² - xy - yz - zx) = 3xyz ..... (iii)

Putting the value of (ii) in (iii),

⇒ xyz (x² + y² + z² - xy - yz - zx) = 3xyz

Cancel xyz both sides -

⇒ (x² + y² + z² - xy - yz - zx) = 3

⇒ x² + y² + z² = 3 + xy + yz + zx

Now, we have to find -

(x - y)² + (y - x)² + (z - x)²

⇒ x² + y² - 2xy + y² + z² - 2yz + z² + x² - 2zx

⇒ 2x² + 2y² + 2z² - 2xy - 2yz - 2zx

⇒ 2 (x² + y² + z²) - 2 (xy + yz + zx)

⇒ 2 (3 + xy + yz + zx) - 2 (xy + yz + zx)

⇒ 6 + 2xy + 2yz + 2zx - 2xy - 2yz - 2zx

⇒ 6

Hence, the answer is 6.