If (x + y + z) = 0 then find the value of \(\frac{{{{\rm{x}}^2}{\rm{\;}} + {\rm{\;}}{{\rm{y}}^2} - {\rm{\;}}{{\rm{z}}^2}}}{{{{({\rm{x\;}} + {\rm{\;y}})}^{2{\rm{\;}}}} - {\rm{\;}}{{\left( {{\rm{x\;}} - {\rm{\;y}}} \right)}^2}{\rm{\;}}}}\)
If (x + y + z) = 0 then find the value of \(\frac{{{{\rm{x}}^2}{\rm{\;}} + {\rm{\;}}{{\rm{y}}^2} - {\rm{\;}}{{\rm{z}}^2}}}{{{{({\rm{x\;}} + {\rm{\;y}})}^{2{\rm{\;}}}} - {\rm{\;}}{{\left( {{\rm{x\;}} - {\rm{\;y}}} \right)}^2}{\rm{\;}}}}\)
1). 1
2). -1/2
3). -1/4
4). 1/2
1 answers
(x + y + z) = 0
⇒ x + y = -z
Squaring both sides
⇒ x2 + y2 + 2xy = z2
⇒ x2 + y2 - z2 = -2xy ---- (1)
Now
(x + y)2 - (x - y)2 = x2 + y2 + 2xy - x2 - y2 + 2xy
⇒ (x + y)2 - (x - y)2 = 4xy ---- (2)
$(\frac{{{{\rm{x}}^2}{\rm{\;}} + {\rm{\;}}{{\rm{y}}^2}{\rm{\;}} - {\rm{\;}}{{\rm{z}}^2}}}{{{{({\rm{x\;}} + {\rm{\;y}})}^{2{\rm{\;}}}} - {\rm{\;}}{{\left( {{\rm{x\;}} - {\rm{\;y}}} \right)}^2}{\rm{\;}}}})$
From equation 1 and 2
$( \Rightarrow {\rm{\;}}\frac{{ - 2{\rm{xy}}}}{{4{\rm{xy\;}}}} = - \frac{1}{2})$
$(\therefore \frac{{{{\rm{x}}^2}{\rm{\;}} + {\rm{\;}}{{\rm{y}}^2}{\rm{\;}} - {\rm{\;}}{{\rm{z}}^2}}}{{{{({\rm{x\;}} + {\rm{\;y}})}^{2{\rm{\;}}}} - {\rm{\;}}{{\left( {{\rm{x\;}} - {\rm{\;y}}} \right)}^2}{\rm{\;}}}} = - \frac{1}{2})$
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