`a`, `b`,`c` are three complex numbers on the unit circle `|z|=1`, such that `abc=a+b+c`. Then `|ab+bc+ca|` is equal to

Let z_1, z_2, z_3 be three complex numbers and a ,b ,c be real numbers not all zero, such that a+b+c=0a n da z_1+b z_2+c z_3=0. Show that z_1, z_2,z_3 are collinear.

If a ,b ,c are nonzero complex numbers of equal moduli and satisfy a z^2+b z+c=0, hen prove that (sqrt(5)-1)//2lt=|z|lt=(sqrt(5)+1)//2.

Let z_1, z_2, z_3 be the three nonzero complex numbers such that z_2!=1,a=|z_1|,b=|z_2|a n d c=|z_3|dot Let |(a, b, c), (b, c, a), (c,a,b) |=0 a r g(z_3)/(z_2) equal to (a) arg((z_3-z_1)/(z_2-z_1))^2 (b) orthocentre of triangle formed by z_1, z_2, z_3, i sz_1+z_2+z_3 (c)if triangle formed by z_1, z_2, z_3 is equilateral, then its area is (3sqrt(3))/2|z_1|^2 (d) if triangle formed by z_1, z_2, z_3 is equilateral, then z_1+z_2+z_3=0

If |z|=1, then the point representing the complex number -1+3z will lie on a. a circle b. a parabola c. a straight line d. a hyperbola

If a,b,c,d are in a geometric sequences, then show that (a-b+c) (b+c+d) = ab + bc + cd.

If a, b and c are distinct positive numbers, then the expression (a + b - c)(b+ c- a)(c+ a -b)- abc is:

If t and c are two complex numbers such that |t|!=|c|,|t|=1a n dz=(a t+b)/(t-c), z=x+i ydot Locus of z is (where a, b are complex numbers) a. line segment b. straight line c. circle d. none of these

Let a ,b ,a n dc be any three nonzero complex number. If |z|=1a n d' z ' satisfies the equation a z^2+b z+c=0, prove that a ( bara) =c (barc) a n d|a||b|=sqrt(a c( bar b )^2)

If |(z-z_1)/(|z-z_2|)=3,w h e r ez_1a n dz_2 are fixed complex numbers and z is a variable complex number, then z lies on a (a).circle with z_1 as its interior point (b).circle with z_2 as its interior point (c).circle with z_1 as its exterior point (d).circle with z_2 as its exterior point

If three positive real numbers a, b, c are in A.P and abc = 4 , then the minimum possible value of b is