In the given figure, if x + y = w + z, then prove that AOB is a line.

If z=z_0+A( bar z -( bar z _0)), w h e r eA is a constant, then prove that locus of z is a straight line.

If x + y + z = 0 , then prove that x^(3)+y^(3)+z^(3)= 3xyz

In the given figure three lines bar(AB) , bar(CD) and bar(EF) intersecting at O. Find the values of x, y and z it is being given that x : y : z = 2 : 3 : 5

If x/ (y+z) = y/(z+x) =z/(x+y) , then prove that the value of each ratio is either 1/2 or -1

Prove that the locus of midpoint of line segment intercepted between real and imaginary axes by the line a z + a z+b=0,w h e r eb is a real parameterand a is a fixed complex number with nondzero real and imaginary parts, is a z+ a z =0.

If (a-x)/(p x)=(a-y)/(q y)=(a-z)/(rz) and p ,q ,and r are in A.P., then prove that x ,y ,z are in H.P.

If z=x+iy and |2z-1| =|z-2|then prove that x^2+y^2=1

If omega!=1 is a cube root of unity and x+y+z!=0, then prove that |[x/(1+omega), y/(omega+omega^2),z/(omega^2+1)],[y/(omega+omega^2),z/(omega^2+1),x/(1+omega)],[(z)/(omega^2+1),x/(1+omega),y/(omega+omega^2)]|=0 if x=y=z

Calculate angleAOC, angleBOD and angleAOE in the adjacent figure given that angleCOD = 90^(@) , angleBOE = 72^(@) and AOB is a straight line,

Given a=x//(y-z), b=y//(z-x), and c=z//(x-y),w h e r e .x , y and z are not all zero, then the value of a b+b c+c a is a. 0 b. 1 c. -1"" d. none of these