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`

If alpha = (z - i)//(z + i) show that, when z lies above the real axis, alpha will lie within the unit circle which has centre at the origin. Find the locus of alpha as z travels on the real axis form -oo "to" + oo

If z!=1 and (z^2)/(z-1) is real, then the point represented by the complex number z lies (1) either on the real axis or on a circle passing through the origin (2) on a circle with centre at the origin (3) either on the real axis or on a circle not passing through the origin (4) on the imaginary axis

If z!=1 and (z^2)/(z-1) is real, then the point represented by the complex number z lies (1) either on the real axis or on a circle passing through the origin (2) on a circle with centre at the origin (3) either on the real axis or on a circle not passing through the origin (4) on the imaginary axis

Find the equation of the circle which lies in the first quadrant and touching each-co ordinate-axis at a distance of 2 units from the origin.

Find the equation to the circles which pass through the origin and cut off intercepts equal to (i) 3 and 4, (ii) 2a and 2b from the x-axis and the y-axis respectively.

A charged point particle having positive charge q , mass m is projected in x-z plane with a speed v_(0) at an angle theta with the x-"axis" , while another identical particle is projected simultaneously along the y-"axis" from the origin (see figure) with an equal speed v_(0) . An electric field vecE= -E_(0)hatk exists in the x-z plane, while a magnetic field vecB=B_(0)hatk exists in the region y gt 0 ( F_(0) , B_(0) gt 0 ). It is observed that both the particles collide after some time. Assume that the space is gravity free. ( i ) Find the time after which the particles collide and the coordinates of the point of collision. ( ii ) Find the velocity v_(0) and the angle theta in terms of the other variables. ( iii ) What is the maximum value of the z- coordinate of the first particle ?

A circular ring of radius R with uniform positive charge density lambda per unit length is located in the y z plane with its center at the origin O. A particle of mass m and positive charge q is projected from that point p( - sqrt(3) R, 0,0) on the negative x - axis directly toward O, with initial speed V. Find the smallest (nonzero) value of the speed such that the particle does not return to P ?

A circular loop of radius R is bent along a diameter and given a shapes as shown in the figure. One of the semicircles (KNM) lies in the x-z plane with their centres and the other one (KLM) in the y-z plane with their centres at the origin. current I is flowing through each of the semi circles as shown in figure. (a) A particle of charge q is released at the origin with a velocity vec(v) = -v_(0)hat(i) . Find the instantaneous force vec(F) on the particle . Assume that space is gravity free. (b) If an external uniform magnetic field B_(0) hat(j) is applied , determine the force vec(F)_(1) and vec(F)_(2) on the semicircles KLM and KNM due to the field and the net force vec(F) on the loop.

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

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