If `x + i y =(a+i b)/(a-i b),`prove that `x^2+y^2=1`.

If x+iy=sqrt((1+i)/(1-i)), prove that x^(2)+y^(2)=1

If x+iy=(1+2i)/(2+i) prove that x^(2)+y^(2)=1

(i) If y = f(x) = (3x-1)/(5x-3) , prove that f(y) = x (ii) If y = f(x) = (ax -b)/(bx-a) , prove that f(y) = x

(i) If the point (x, y) is equidistant from the points (a+b, b-a) and (a-b, a+b), prove that bx = ay. (ii) If the distances of P(x, y) from A(5, 1) and B(-1, 5) are equal then prove that 3x =2y.

If (x+iy)(p+iq)=(x^(2)+y^(2))i, prove that x=q,y=p

if (1+i x/a) (1+i x/b)(1+i x/c)…=A+iB then prove that (1+ x^2/a^2)(1+x^2/b^2)(1+x^2/c^2)…=A^2+B^2

If x=a sec theta,y=b tan theta, then prove that (x^(2))/(a^(2))-(y^(2))/(b^(2))=1