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

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

If x+iy=sqrt((1+i)/(1-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

A triangle has vertices A_i(x_i , y_i)fori=1,2,3 If the orthocentre of triangle is (0,0), then prove that |x_2-x_3y_2-y_3y_1(y_2-y_3)+x_1(x_2-x_3)x_3-x_1y_2-y_3y_2(y_3-y_1)+x_1(x_3-x_1)x_1-x_2y_2-y_3y_3(y_1-y_2)+x_1(x_1-x_2)|=0

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 the normal at four points P_(i)(x_(i), (y_(i)) l, I = 1, 2, 3, 4 on the rectangular hyperbola xy = c^(2) meet at the point Q(h, k), prove that x_(1) + x_(2) + x_(3) + x_(4) = h, y_(1) + y_(2) + y_(3) + y_(4) = k x_(1)x_(2)x_(3)x_(4) =y_(1)y_(2)y_(3)y_(4) =-c^(4)