Let`A(z_1), B(z_2) and C(z_3)` forms an acute angled triangle in argand plane having origin as it’s orthocentre. Prove that
`z_1barz_2+bar(z)_1 z_2=z_2barz_3+barz_2z_3=z_3barz_1 +barz_3z_1`

Let vertices of an acute-angled triangle are A(z_(1)),B(z_(2)), and C(z_(3))* If the origin O is he orthocentre of the triangle,then prove that z_(1)(bar(z))_(2)+(bar(z))_(1)z_(2)=z_(2)(bar(z))_(3)+(bar(z))_(2)z_(3)=z_(3)(bar(z))_(1)+(bar(z))_(3)z_(1)

Let A(z_1),B(z_2) and C(z_3) be the vertices of an equilateral triangle in the Argand plane such that |z_1|=|z_2|=|z_3|. Then (A) (z_2+z_3)/(2z_1-z_2-z_3) is purely real (B) (z_2-z_3)/(2z_1-z_2-z_3) is purely imaginary (C) |arg(z_1/z_2)|=2 arg((z_3-z_2)/(z_1-z_2))| (D) none of these

Let A(z_(1)),B(z_(2)),C(z_(3)) be the vertices of an equilateral triangle ABC in the Argand plane, then the number (z_(2)-z_(3))/(2z_(1)-z_(2)-z_(3)) , is

Solve: z +2barz=ibarz

Prove that |(z_1, z_2)/(1-barz_1z_2)|lt1 if |z_1|lt1,|z_2|lt1

If z_(1) = 3+ 2i and z_(2) = 2-i then verify that (i) bar(z_(1) + z_(2)) = barz_(1) + barz_(2)

Show that the equation of the circle in the complex plane with z_1 & z_2 as its diameter can be expressed as , 2z barz-(barz_1+barz_2)z-(barz_1+barz_2)barz+z_1barz_2+z_2barz_1=0

if z=3 -2i, then verify that (i) z + barz = 2Rez (ii) z - barz = 2ilm z

Prove that |1-barz_1z_2|^2-|z_1-z_2|^2=(1-|z_1|^2)(1-|z_2|^2) .

If z_(1),z_(2)andz_(3) are the affixes of the vertices of a triangle having its circumcentre at the origin. If zis the affix of its orthocentre, prove that Z_(1)+Z_(2)+Z_(3)-Z=0.