If x - y + z = 0, then find the value of $$\frac{{{y^2}}}{{2xz}} - \frac{{{x^2}}}{{2yz}} - \frac{{{z^2}}}{{2xy}}?$$

Solution (By Examveda Team)

$$\eqalign{ & \frac{{{y^2}}}{{2xz}} - \frac{{{x^2}}}{{2yz}} - \frac{{{z^2}}}{{2xy}} \cr & \frac{{{y^3} - {x^3} - {z^3}}}{{2xyz}}\,......\,\left( 1 \right) \cr & x - y + z = 0 \cr & x + z = y \cr & {\text{Cubing both side}} \cr & {\left( {x + z} \right)^3} = {y^3} \cr & {x^3} + {z^3} + 3\left( {x + z} \right)\left( x \right)\left( z \right) = {y^3} \cr & {x^3} + {z^3} + 3\left( y \right)\left( x \right)\left( z \right) = {y^3} \cr & 3xyz = {y^3} - {x^3} - {z^3} \cr & {\text{Put in equation }}\left( 1 \right) \cr & \frac{{3xyz}}{{2xyz}} = \boxed{\frac{3}{2}} \cr} $$