3If we interchange x and y in a given eqn the eqn doesnt change,so i think these type of eqns are symmetrical with respect to x and y.
We say x,y,z can be symmetric in an expression..........pls explain wht do we mean by saying so ?
To illustrate my doubt, I'm posting one example :
Case I :
Let x,y,z be real variables which satisfy xy+yz+zx=7 and x+y+z=6
We find the range in which x lies between 6-2√153 and 6+2√153 and we say , "since x,y,z are symmetrically placed, all of them lie within the same limits. Why is it so ?
Case II:
For x2+a2=8x+6a, x and a have different limits unlike Case I.........why ?
Please help.
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3 Answers
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66We say that an expression f(x_1,x_2,\ldots, x_n) in n variables is symmetrical w.r.t. these variables, if interchanging any pair of them leave the value of the expression unchanged.
For instance, the expression f(x,y,z)=xy+yz+zx is symmetric w.r.t. x,y,z. You should note that the expression
g(x,y,z)=A(x+y+z)
is the only symmetric function in three variables of degree 1. (Here A is some constant)
Similarly,
g_1(x,y,z)=A(x^2+y^2+z^2) and g_2(x,y,z)=A(xy+yz+zx) are the only symmetrical functions in three variables of degree 2.
As for the examples you gave, probably now you can answer both your queries.
