Complex Numbers

Find the centre and radius of circle given by \(\,\left| {{{z - \alpha } \over {z - \beta }}} \right| = k,k \ne 1\,\)
where, $${\rm{z = x + iy, }}\alpha {\rm{ = }}\,{\alpha _1}{\rm{ + i}}{\alpha _2}{\rm{,}}\,\beta = {\beta _1}{\rm{ ...

If \(\,\left| z \right| = 1\) and \(\omega = {{z - 1} \over {z + 1}}\) (where \(z \ne - 1\)), then \({\mathop{\rm Re}\nolimits} \left( \omega \right)\) is

If \({z_1}\) and \({z_2}\) are two complex numbers such that \(\,\left| {{z_1}} \right| < 1 < \left| {{z_2}} \right|\,\) then prove that \(\,\left| {{{1 - {z_1}\overline {{z_2}} } \over {{z_1} - {z_2}}}} \right| < 1\).

Prove that there exists no complex number z such that \(\left| z \right| < {1 \over 3}\,and\,\sum\limits_{r = 1}^n {{a_r}{z^r}} = 1\) where \(\left| {{a_r}} \right| < 2\).

For all complex numbers \({z_1},\,{z_2}\) satisfying \(\left| {{z_1}} \right| = 12\) and \(\left| {{z_2} - 3 - 4i} \right| = 5,\)
the minimum value of \(\left| {{z_1} - {z_2}} \right|\) is

Let a complex number \(\alpha ,\,\alpha \ne 1\), be a root of the equation \({z^{p + q}} - {z^p} - {z^q} + 1 = 0\), where p, q are distinct primes. Show that either $$1 + \alpha + {\alpha ^2} + .... + {\alpha ^{p - 1}} = 0\,or\,1 + \alpha...

Let \(\omega\) \(= - {1 \over 2} + i{{\sqrt 3 } \over 2},\) then the value of the det.
\(\,\left| {\matrix{ 1 & 1 & 1 \cr 1 & { - 1 - {\omega ^2}} & {{\omega ^2}} \cr 1 & {{\omega ^2}} & {{\omega ^4}} \cr } } \right|\) ...

The complex numbers \({z_1},\,{z_2}\) and \({z_3}\) satisfying \({{{z_1} - {z_3}} \over {{z_2} - {z_3}}} = {{1 - i\sqrt 3 } \over 2}\,\) are the vertices of a triangle which is

Let \({z_1}\) and \({z_2}\) be \({n^{th}}\) roots of unity which subtend a right angle at the origin. Then \(n\) must be of the form

If \({z_1},\,{z_2}\) and \({z_3}\) are complex numbers such that \(\left| {{z_1}} \right| = \left| {{z_2}} \right| = \left| {{z_3}} \right| = \left| {{1 \over {{z_1}}} + {1 \over {{z_2}}} + {1 \over {{z_3}}}} \right| = 1,\) then $$\left| {{...

If \(\arg \left( z \right) < 0,\) then \(\arg \left( { - z} \right) - \arg \left( z \right) =\)

\(If\,i = \sqrt { - 1} ,\,\,then\,\,4 + 5{\left( { - {1 \over 2} + {{i\sqrt 3 } \over 2}} \right)^{334}} + 3{\left( { - {1 \over 2} + {{i\sqrt 3 } \over 2}} \right)^{365}}\) is equal to

For complex numbers z and w, prove that \({\left| z \right|^2}w - {\left| w \right|^2}z = z - w\) if and only if \(z = w\,or\,z\overline {\,w} = 1\).

If \({\omega}\) is an imaginary cube root of unity, then \({(1\, + \omega \, - {\omega ^2})^7}\) equals

If \(\,\left| {\matrix{ {6i} & { - 3i} & 1 \cr 4 & {3i} & { - 1} \cr {20} & 3 & i \cr } } \right| = x + iy\) , then

The value of the sum \(\,\,\sum\limits_{n = 1}^{13} {({i^n}} + {i^{n + 1}})\) , where i = \(\sqrt { - 1}\), equals

Let \({z_1}\) and \({z_2}\) be roots of the equation \({z^2} + pz + q = 0\,\) , where the coefficients p and q may be complex numbers. Let A and B represent \({z_1}\) and \({z_2}\) in the complex plane. If \(\angle AOB = \alpha \ne 0\,\) a...

For positive integers \({n_1},\,{n_2}\) the value of the expression \({\left( {1 + i} \right)^{^{{n_1}}}} + {\left( {1 + {i^3}} \right)^{{n_1}}} + {\left( {1 + {i^5}} \right)^{{n_2}}} + {\left( {1 + {i^7}} \right)^{{n_2}}},\)
where $$i = \...

Find all non-zero complex numbers Z satisfying \(\overline Z = i{Z^2}\).

The value of the expression
$$1 \bullet \left( {2 - \omega } \right)\left( {2 - {\omega ^2}} \right) + 2 \bullet \left( {3 - \omega } \right)\left( {3 - {\omega ^2}} \right) + \,....... + \left( {n - 1} \right).\left( {n - \omega } \right)\...

Let \(z\) and \(\omega\) be two complex numbers such that
\(\left| z \right| \le 1,\) \(\left| \omega \right| \le 1\) and \(\left| {z + i\omega } \right| = \left| {z - i\overline \omega } \right| = 2\) then \(z\) equals

Let \(z\) and \(\omega\) be two non zero complex numbers such that
\(\left| z \right| = \left| \omega \right|\) and \({\rm A}rg\,z + {\rm A}rg\,\omega = \pi ,\) then \(z\) equals

If \(\omega \,\left( { \ne 1} \right)\) is a cube root of unity and \({\left( {1 + \omega } \right)^7} = A + B\,\omega\) then \(A\) and \(B\) are respectively

If \(\left| {Z - W} \right| \le 1,\left| W \right| \le 1\), show that \({\left| {Z - W} \right|^2} \le {(\left| Z \right| - \left| W \right|)^2} + {(ArgZ - Arg\,W)^2}\)

If \(i{z^3} + {z^2} - z + i = 0\) , then show that \(\left| z \right| = 1\).

Suppose Z1, Z2, Z3 are the vertices of an equilateral triangle inscribed in the circle \(\left| Z \right| = 2.\) If Z1 = \(1 + i\sqrt 3\) then Z2 = ......., Z3 =..............

\(ABCD\) is a rhombus. Its diagonals \(AC\) and \(BD\) intersect at the point \(M\) and satisfy \(BD\) = 2\(AC\). If the points \(D\) and \(M\) represent the complex numbers \(1 + i\) and \(2 - i\) respectively, then A represents the comp[l...

\({\rm{z }} \ne {\rm{0}}\) is a complex number
Column I
(A) Re z = 0
(B) Arg \(z = {\pi \over 4}\)
Column II
(p) Re\({z^2}\) = 0
(q) Im\({z^2}\) = 0
(r) Re\({z^2}\) = Im\({z^2}\)

Let \({z_1}\) = 10 + 6i and \({z_2}\) = 4 + 6i. If Z is any complex number such that the argument of \({{(z - {z_1})} \over {(z - {z_2})}}\,is{\pi \over 4}\) , then prove that \(\left| {z - 7 - 9i} \right| = 3\sqrt 2\).

If \(a,\,b,\,c,\) are the numbers between 0 and 1 such that the ponts \({z_1} = a + i,{z_2} = 1 + bi\) and \({z_3} = 0\) form an equilateral triangle,
then a= .......and b=..........

The cube roots of unity when represented on Argand diagram form the vertices of an equilateral triangle.

For any two complex numbers \({z_1},{z_2}\) and any real number a and b.
\(\,{\left| {a{z_1} - b{z_2}} \right|^2} + {\left| {b{z_1} + a{z_2}} \right|^2} = .........\)

The value of \(\sum\limits_{k = 1}^6 {(\sin {{2\pi k} \over 7}} - i\,\cos \,{{2\pi k} \over 7})\) is

If \({{{z_1}}}\) and \({{{z_2}}}\) are two nonzero complex numbers such that \(\left| {{z_1}\, + {z_2}} \right| = \left| {{z_1}} \right|\, + \left| {{z_2}} \right|\,\), then Arg \({z_1}\) - Arg \({z_2}\) is equal to

If the expression
\(${{\left[ {\sin \left( {{x \over 2}} \right) + \cos {x \over 2} + i\,\tan \left( x \right)} \right]} \over {\left[ {1 + 2\,i\,\sin \left( {{x \over 2}} \right)} \right]}}\)$
is real, then the set of all possible values o...

Let \({z_1}\) and \({z_2}\) be complex numbers such that \({z_1}\) \(\ne\) \({z_2}\) and \(\left| {{z_1}} \right| =\,\left| {{z_2}} \right|\). If \({z_1}\) has positive real and \({z_2}\) has negative imaginary part, then $${{{z_1}\, ...

Show that the area of the triangle on the Argand diagram formed by the complex numbers z, iz and z + iz is \({1 \over 2}\,{\left| z \right|^2}\) .

If \({z_1}\) = a + ib and \({z_2}\) = c + id are complex numbers such that \(\left| {{z_1}} \right| = \left| {{z_2}} \right| = 1\) and \({\mathop{\rm Re}\nolimits} ({z_1}\,{\overline z _2}) = 0\), then the pair of complex numbers \({w_1}\)...

If \(a,\,b,\,c\) and \(u,\,v,\,w\) are complex numbers representing the vertics of two triangles such that \(c = \left( {1 - r} \right)a + rb\) and \(w = \left( {1 - r} \right)u + rv,\) where \(w = \left( {1 - r} \right)u + rv,\) is a compl...

If three complex numbers are in A.P. then they lie on a circle in the complex plane.

If 1, \({{a_1}}\), \({{a_2}}\)......,\({a_{n - 1}}\) are the n roots of unity, then show that (1- \({{a_1}}\)) (1- \({{a_2}}\)) (1- \({{a_3}}\)) ....\((1 - \,a{ - _{n - 1}}) = n\)

If the complex numbers, \({Z_1},{Z_2}\) and \({Z_3}\) represent the vertics of an equilateral triangle such that
\(\left| {{Z_1}} \right| = \left| {{Z_2}} \right| = \left| {{Z_3}} \right|\) then \({Z_1} + {Z_2} + {Z_3} = 0.\)

The points z1, z2, z3, z4 in the complex plane are the vertices of a parallelogram taken in order if and only if

If \(z = x + iy\) and \(\omega = \left( {1 - iz} \right)/\left( {z - i} \right),\) then \(\,\left| \omega \right| = 1\) implies that, in the complex plane,

Prove that the complex numbers \({{z_1}}\), \({{z_2}}\) and the origin form an equilateral triangle only if \(z_1^2 + z_2^2 - {z_1}\,{z_2} = 0\).

The inequality |z-4| < |z-2| represents the region given by

If \(z = {\left( {{{\sqrt 3 } \over 2} + {i \over 2}} \right)^5} + {\left( {{{\sqrt 3 } \over 2} - {i \over 2}} \right)^5},\) then

For complex number \({z_1} = {x_1} + i{y_1}\) and \({z_2} = {x_2} + i{y_2},\) we write \({z_1} \cap {z_2},\,\,if\,\,{x_1} \le {x_2}\,\,and\,\,{y_1} \le {y_2}.\)
Then for all complex numbers \(z\,\,with\,\,1 \cap z,\) we have $${{1 - z} \ov...

The complex numbers \(z = x + iy\) which satisfy the equation \(\,\left| {{{z - 5i} \over {z + 5i}}} \right| = 1\) lie on

Let the complex number \({{z_1}}\), \({{z_2}}\) and \({{z_3}}\) be the vertices of an equilateral triangle. Let \({{z_0}}\) be the circumcentre of the triangle. Then prove that \(z_1^2 + z_2^2 + z_3^2 = 3z_0^2\).