Given `(x_(1) + iy_(1)) (x_(2) + iy_(2))…(x_(n) + iy_(n))` = a + ib, show that
`(x_(1)^(2)+y_(1)^(2))(x_(2)^(2) + y_(2)^(2))(x_(3)^(2) + y_(3)^(2))…(x_(n)^(2) + y_(n)^(2)) = a^(2) + b^(2)`

Given (x_(1) + iy_(1)) (x_(2) + iy_(2))…(x_(n) + iy_(n)) = a + ib, show that sum_(r=1)^(n) tan^(-1)""(y_(r))/(x_(r))= tan^(-1)((b)/(a))+2k pi, " k " in ZZ

If y= (tan^(-1)x)^(2) , show that (x^(2)+1)^(2) y_(2)+2x(x^(2)+1)y_(1)=2 .

Three points P (h, k), Q(x_(1) , y_(1))" and " R (x_(2) , Y_(2)) lie on a line. Show that (h - x_(1)) (y_(2) - y_(1)) = (k - y_(1)) (x_(2) - x_(1)) .

If (x+iy)^(3)= u +iv , then show that (u)/(x)+(v)/(y)=4(x^(2)-y^(2)) .

If x-iy= sqrt((a-ib)/(c-id)) , prove that x^(2)+y^(2)=(a^(2)+b^(2))/(c^(2)+d^(2)) .

it x_(1)^(2) +2y_(1)^(2)+3z_(1)^(2)=x_(2)^(2)+2y_(2)^(2)+3z_(2)^(2)=x_(3)^(2)+2y_(3)^(2)+3z_(3)^(2)=2 " and " x_(2)x_(3) +2y_(2)y_(3)+3z_(2)z_(3)=x_(3)x_(1)+2y_(3)y_(1)+3z_(3)z_(1)=x_(1)x_(2)+2y_(1)y_(2)+3z_(1)z_(2)=1 Then find the value of |{:(x_(1),,y_(1),,z_(1)),(x_(2),,y_(2),,z_(2)),(x_(3),,y_(3),,z_(3)):}|

If x+iy = (a+ib)/(a-ib) , prove that x^(2) + y^(2)=1 .

If (cos theta + isin theta)^(2) = x + iy, then show that x^(2) + y^(2) = 1

Find the equation of a line through the given pair of points (x_(1), y_(1)), (x_(2), y_(2)) (2, (2)/(3)) and ((-1)/(2), -2)

A triangle has vertices A_(i) (x_(i),y_(i)) for i= 1,2,3,. If the orthocenter of triangle is (0,0) then prove that |{:(x_(2)-x_(3),,y_(2)-y_(3),,y_(1)(y_(2)-y_(3))+x_(1)(x_(2)-x_(3))),(x_(3)-x_(1) ,,y_(3)-y_(1),,y_(2)(y_(3)-y_(1))+x_(2)(x_(3)-x_(1))),( x_(1)-x_(2),,y_(1)-y_(2),,y_(3)(y_(1)-y_(2))+x_(3)(x_(1)-x_(2))):}|=0