If `alpha=(z-i)/(z+i),` show that, when `z` lies above the real axis, `alpha` will lie within the unit circle which has center at the origin. Find the locus of `alphaa sz` travels on the real axis from `-ooto+oodot`

Find the range of real number alpha for which the equation z+alpha|z-1|+2i=0 has a solution.

Let z=x+iy and omega=(1-iz)/(z-i). If |omega|=1 show the in the complex plane the point z lies on the real axis.

A circle with center at the origin and radius equal to a meets the axis of x at Aa n dB. P(alpha) and Q(beta) are two points on the circle so that alpha-beta=2gamma , where gamma is a constant. Find the locus of the point of intersection of A P and B Qdot

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

If the argument of (z-a)(barz-b) is equal to that ((sqrt(3)+i)(1+sqrt(3)i)/(1+i)) where a,b,c are two real number and z is the complex conjugate o the complex number z find the locus of z in the rgand diagram. Find the value of a and b so that locus becomes a circle having its centre at 1/2(3+i)

C_1 and C_2 are circle of unit radius with centers at (0, 0) and (1, 0), respectively, C_3 is a circle of unit radius. It passes through the centers of the circles C_1a n dC_2 and has its center above the x-axis. Find the equation of the common tangent to C_1a n dC_3 which does not pass through C_2dot

A square of side 'a' lies above the x-axis and has one vertex at the origin. The side passing through the origin makes an angle alpha (0 < alpha' < pi /4 ) with the positive direction of x-axis. Find the equation of diagonal not passing through the origin ?

Find the points on the z-axis which are at a distance of sqrt(29) unit from the point ( 2, -3 , -2) .

A long round conductor of cross-sectional area A is made of a material whose resistivity depends only on the distance r from the axis of the conductor as p=alpha/r^2 where a is alpha constant. Find (a) the resistance per unit length of such a conductor, (b) the electric field strength in the conductor due to which a current I flows through it.