Prove that the locus of midpoint of line segment intercepted between real and imaginary axes by the line `bara z + a bar 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+ bara barz =0.`

If one root of the equation z^2-a z+a-1=0i s(1+i),w h e r ea is a complex number then find the root.

Complex numbers whose real and imaginary parts x and y are integers and satisfy the equation 3x^(2)-|xy|-2y^(2)+7=0

Show that the equation of a circle passing through the origin and having intercepts aa n db on real and imaginary axes, respectively, on the argand plane is given by z bar z =a a(R e z)+b(I mz)dot

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.

Let z=9+b i ,w h e r eb is nonzero real and i^2=-1. If the imaginary part of z^2a n dz^3 are equal, then b/3 is ______.

A curve is such that the mid-point of the portion of the tangent intercepted between the point where the tangent is drawn and the point where the tangent meets the y-axis lies on the line y=xdot If the curve passes through (1,0), then the curve is (a) ( b ) (c)2y=( d ) x^(( e )2( f ))( g )-x (h) (i) (b) ( j ) (k) y=( l ) x^(( m )2( n ))( o )-x (p) (q) (c) ( d ) (e) y=x-( f ) x^(( g )2( h ))( i ) (j) (k) (d) ( l ) (m) y=2(( n ) (o) x-( p ) x^(( q )2( r ))( s ) (t))( u ) (v)

The locus of point z satisfying R e(1/z)=k ,w h e r ek is a nonzero real number, is a. a straight line b. a circle c. an ellipse d. a hyperbola

Let z be a complex number such that the imaginary part of z is nonzero and a = z^2 + z + 1 is real. Then a cannot take the value

If z!=0 is a complex number, then prove that R e(z)=0 rArr Im(z^2)=0.

If z=|-5 3+4i5-7i3-4i6 8+7i5+7i8-7i9|,t h e nz is purely real purely imaginary a+i b ,w h e r ea!=0,b!=0 d. a+i b ,w h e r eb=4