Show that the triangle whose vertices are `z_(1)z_(2)z_(3)andz_(1)'z_(2)'z_(3)'` are directly similar , if `|{:(z_(1),z'_(1),1),(z_(2),z'_(2),1),(z_(3),z'_(3),1):}|=0`

Find the co-ordinates of the centroid of the triangle whose vertices are (x_(1),y_(1),z_(1)), (x_(2),y_(1),z_(2)) and (x_(3),y_(3),z_(3)) .

If |z_(1)|=|z_(2)|andamp(z_(1))+amp(z_(2))=0, then

If |z_(1)|=2,|z_(2)|=3,|z_(3)|=4and|z_(1)+z_(2)+z_(3)|=5. "then"|4z_(2)z_(3)+9z_(3)z_(1)+16z_(1)z_(2)| is

Let z_(1) and z_(2) be two distinct complex numbers and z=(1-t)z_(1)+tz_(2) , for some real number t with 0 lt t lt 1 and i=sqrt(-1) . If arg(w) denotes the principal argument of a non-zero complex number w, then a. abs(z-z_(1))+abs(z-z_(2))=abs(z_(1)-z_(2)) b. arg(z-z_(1))=arg(z-z_(2)) c. |{:(z-z_(1),bar(z)-bar(z)_(1)),(z_(2)-z_(1),bar(z)_(2)-bar(z)_(1)):}|=0 d. arg(z-z_(1))=arg(z_(2)-z_(1))

bb"statement-1" " Let " z_(1),z_(2) " and " z_(3) be htree complex numbers, such that abs(3z_(1)+1)=abs(3z_(2)+1)=abs(3z_(3)+1) " and " 1+z_(1)+z_(2)+z_(3)=0, " then " z_(1),z_(2),z_(3) will represent vertices of an equilateral triangle on the complex plane. bb"statement-2" z_(1),z_(2),z_(3) represent vertices of an triangle, if z_(1)^(2)+z_(2)^(2)+z_(3)^(2)+z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1)=0

Find the circumstance of the triangle whose vertices are given by the complex numbers z_(1),z_(2) and z_(3) .

If z_(1), z_(2) are 1-i, -2 + 4i respectively, find I_(m)((z_(1)z_(2))/(bar(z_(1)))) .

Let z_(1),z_(2) and z_(3) be three non-zero complex numbers and z_(1) ne z_(2) . If |{:(abs(z_(1)),abs(z_(2)),abs(z_(3))),(abs(z_(2)),abs(z_(3)),abs(z_(1))),(abs(z_(3)),abs(z_(1)),abs(z_(2))):}|=0 , prove that (i) z_(1),z_(2),z_(3) lie on a circle with the centre at origin. (ii) arg(z_(3)/z_(2))=arg((z_(3)-z_(1))/(z_(2)-z_(1)))^(2) .

If arg (z_(1))=(17pi)/18 and arg (z_(2))=(7pi)/18, find the principal argument of z_(1)z_(2) and (z_(1)//z_(2)).

If z_(1),z_(2),z_(3) andz_(4) are the roots of the equation z^(4)=1, the value of sum_(i=1)^(4)z_i^(3) is