If `z_1=x_1+iy_1,z_2=x_2+iy_2 and z_1 = (i(z_2+1))/(z_2-1)`, prove that `x_1^2+y_1^2-x_1= (x_2^2+y_2^2+2x_2-2y_2+1)/((x_2-1)^2+y_2^2)`

If a x_1^2+b y_1^2+c z_1^2=a x_2 ^2+b y_2 ^2+c z_2 ^2=a x_3 ^2+b y_3 ^2+c z_3 ^2=d ,a x_2 x_3+b y_2y_3+c z_2z_3=a x_3x_1+b y_3y_1+c z_3z_1=a x_1x_2+b y_1y_2+c z_1z_2=f, then prove that |(x_1, y_1, z_1), (x_2, y_2, z_2), (x_3,y_3,z_3)|=(d-f){((d+2f))/(a b c)}^(1//2)

If z= x+ yi and (|z-1 -i|+4)/(3|z-1-i|-2)=1 , show that x^(2) + y^(2)- 2x-2y-7=0

If z= x +yi and (|z-1-i|+4)/(3|z-1-i|-2)=1 , show that x^(2) + y^(2) -2x-2y- 7=0

If z=x+ iy such that the argument of (z-1)/(z+1) is always pi/4 . Prove that x^2 + y^2-2y=1

If a x1 2+b y1 2+c z1 2=a x1 2+b y2 2+c z2 2=a x3 2+b y3 2+c z3 2=d\ a n d\ a x_2x_3+b y_2y_3+c z_2z_3=a x_3x_1+b y_3y_1+c z_3z_1=a x_1x_2+b y_1y_2+c z_1z_2=f then prove that |x_1y_1z_1x_2y_2z_1x_3y_3z_3|=(d-f)[(d+2f)/(a b c)]^(//2)(a , b ,\ c!=0)

If cos^(-1)x+cos^(-1)y+cos^(-1)z=pi , prove that x^2+y^2+z^2+2x y z=1.

If cos^(-1)x+cos^(-1)y+cos^(-1)z=pi , prove that x^2+y^2+z^2+2x y z=1.

If x y+y z+x z=1 ,then prove that x/(1-x^2)+y/(1-y^2)+z/(1-z^2)=(4x y z)/((1-x^2)(1-y^2)(1-z^2)

If x y+y z+x z=1 ,then prove that x/(1-x^2)+y/(1-y^2)+z/(1-z^2)=(4x y z)/((1-x^2)(1-y^2)(1-z^2)

For x_(1), x_(2), y_(1), y_(2) in R if 0 lt x_(1)lt x_(2)lt y_(1) = y_(2) and z_(1) = x_(1) + i y_(1), z_(2) = x_(2)+ iy_(2) and z_(3) = (z_(1) + z_(2))//2, then z_(1) , z_(2) , z_(3) satisfy :