1.

Giventhat thesystem of equationsx=cy+bz ,y=az+cx , z=bx +ay has nonzerosolutions andand atleastone of the a,b,c is a properfraction. Systemhas solution such that

Answer»

`X,y,z -= (1-2a^(2)):(1-2b^(2)):(1-2c^(2))`
`x.y.z -= (1)/(1-2a^(2)):(1)/(1-2b^(2)):(1)/(1-2c^(2))`
`x.y.z -= (a)/(1-a^(2)):(b)/(1-b^(2)):( c)/(1-c^(2))`
`x.y.z -= sqrt(1-a^(2)):sqrt(1-b^(2)):sqrt(1-c^(2))`

Solution :Thesystemof EQUATION
`-x+cy+bz=0`
`cx-y+az=0`
`bx+ay-z=0`
has a nonzerosolution if
`Delta = |{:(-1,,c,,b),(c,,-1,,a),(b,,a,,-1):}|=0`
then CLEARLY the SYSTEMHAS infinitely manysolutions .From(1) and (2) we have
`(x)/(ac+b) =(y)/(bc+a)=(z)/(1-c^(2)) `
`" or" (x^(2))/((1-a^(2))(1-c^(2)))=(y^(2))/((1-b^(2))(1-c^(2)a)) =(z^(2))/((1-c^(2))^(2))`[From (4)]
`"or" (x^(2))/(1-a^(2))=(y^(2))/(1-b^(2)) =(z^(2))/(1-c^(2))`
from (5)we see that `1-a^(2),1-b^(2),1-c^(2)` are allpositiveor allnegative .Giventhat oneof a,b,cis properfraction so
`1-a^(2) gt ,1-b^(2) gt 0,1-c^(2) gt 0` WHICHGIVES `a^(2) +b^(2)+c^(2) lt 3`
using(4) and (6) we get
`1lt 3+2 abc`
`"or" abc gt -1`


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