Let x, y, and z be nonzero real numbers such that xy + xz + yz = 0 and x + y + z \( \neq\) 0. Find all possible values of \(\frac{1}{x^2 - yz} + \frac{1}{y^2 - xz} + \frac{1}{z^2 - xy}.\)
xy + xz + yz = 0 (1)
yz = -(xy + xz) ⇒ -yz = +xy + xz (2)
xz = - (xy + yz) ⇒ -xz = + xy + yz (3)
xy = - (xz + yz) ⇒ -xy = + xz + yz (4)
1 1 1
______ + ________ + ________ =
x^2-yz y^2 - xz z^2 - xy
[ sub in (2), (3) and (4) in the denominators ]
1 1 1
___________ + _____________ + _____________ =
x^2 + xy + xz y^2 + xy + yz z^2 + xz + yz
1 1 1
__________ + ___________ + _____________ =
x ( x + y + z) y ( x + y + z) z(x + y + z)
yz xz xy
____________ + ____________ + ____________ =
xyz(x + y + z) xyz(x + y + z) xyz ( x+ y + z)
( xy + xz + yz )
________________ =
xyz ( x + y + z)
sub in (1) for the numerator
( 0 )
______________ = 0
xyz ( x + y + z)