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Question

Let a,b,c be any real numbers.Suppose that there are real numbers x,y,z not all zero such that #x=cy+bz ,y=az+cx and z=bx+ay# then find the value of #a^2+b^2+c^2#.

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Answer 1 · 2018-06-04T19:05:31

# 1-2abc#.

We first eliminate #z#.

#x=cy+bz," where, "z=bx+ay#

#:. x=cy+b(bx+ay)=cy+b^2x+aby#.

# :. x(1-b^2)=y(c+ab)#

#:. x/y=(c+ab)/(1-b^2)...........................(1)#.

Similarly, #y=az+cx=a(bx+ay)+cx=abx+a^2y+cx#,

#:. y(1-a^2)=x(c+ab)#.

#:. x/y=(1-a^2)/(c+ab)............................(2)#.

# (1) and (2) rArr (c+ab)/(1-b^2)=x/y=(1-a^2)/(c+ab)#,

# i.e., (c+ab)^2=(1-a^2)(1-b^2)#.

#:. c^2+2abc+a^2b^2=1-a^2-b^2+a^2b^2#.

#:. a^2+b^2+c^2=1-2abc#, is the desired value!