If a,b,` x in R` and `(1-ix)/(1+ix)=a+ib`,then prove that `a^2+b^2=1`

If a,b, xin R and if (1-ix)/(1+ix)=a-ib ,prove that a^2 + b^2 = 1 .

If a,b,c are in G.P and a^(1/x)=b^(1/y)=c^(1/z) then prove that x,y,z are in A.P.

If a,b,c are positive unequal real numbers then prove that a^2+b^2+c^2=1=x^2+y^2+z^2 ,then prove that az+bx+cyle1 .

If log((a+2b)/4)=1/2(loga+logb) ,then prove that a^2+4b^2=12ab

if (x+iy)^(1//3)=a+ib ,show that x/a+y/b=4(a^2-b^2)

If G.M. of a and b is (a^(n+1)+b^(n+1))/(a^n+b^n) ,then prove that n=-1/2 .

If G.M. of a and b is (a^n+b^n)/(a^(n-1)+b^(n-1)) then prove that n=1/2 .

If a + b + c = 0 , then prove that 1/(2a^2+bc)+1/(2b^2+ca)+1/(2c^2+ab)=0

If a + b + c = 0 , then prove that (a^2+b^2+c^2)/(a^3+b^3+c^3)+2/3(1/a+1/b+1/c)=0

If logp/(b-c)=logq/(c-a)=logr/(a-b) ,then prove that p^a.q^b.r^c=1