Advertisements
Advertisements
प्रश्न
If \[\left| z + 1 \right| = z + 2\left( 1 + i \right)\],find z.
उत्तर
Let \[z = x + iy\]
Then,
\[z + 1 = \left( x + 1 \right) + iy\]
\[ \Rightarrow \left| z + 1 \right| = \sqrt{\left( x + 1 \right)^2 + y^2}\]
\[\therefore \left| z + 1 \right| = z + 2\left( 1 + i \right)\]
\[ \Rightarrow \sqrt{x^2 + 2x + 1 + y^2} = \left( x + iy \right) + 2 + 2i\]
\[ \Rightarrow \sqrt{x^2 + 2x + 1 + y^2} = \left( x + 2 \right) + i\left( y + 2 \right)\]
\[ \Rightarrow \sqrt{x^2 + 2x + 1 + y^2} = \left( x + 2 \right) \text { and } y + 2 = 0\]
\[ \Rightarrow x^2 + 2x + 1 + y^2 = \left( x + 2 \right)^2 \text { and } y = - 2\]
\[ \Rightarrow x^2 + 2x + 1 + y^2 = x^2 + 4x + 4 \text { and } y = - 2\]
\[ \Rightarrow y^2 = 2x + 3 \text { and } y = - 2\]
\[ \Rightarrow 4 = 2x + 3 \text { and } y = - 2\]
\[ \Rightarrow 2x = 1 \text { and } y = - 2\]
\[ \Rightarrow x = \frac{1}{2} \text { and } y = - 2\]
\[\therefore z = x + iy = \frac{1}{2} - 2i\]
Thus,
\[z = \frac{1}{2} - 2i\]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: i9 + i19
Express the given complex number in the form a + ib: 3(7 + i7) + i(7 + i7)
Evaluate the following:
i457
Evaluate the following:
\[i^{37} + \frac{1}{i^{67}}\].
Express the following complex number in the standard form a + i b:
\[\frac{1}{(2 + i )^2}\]
Express the following complex number in the standard form a + i b:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
Find the real value of x and y, if
\[(1 + i)(x + iy) = 2 - 5i\]
Find the multiplicative inverse of the following complex number:
1 − i
Find the multiplicative inverse of the following complex number:
\[(1 + i\sqrt{3} )^2\]
If \[z_1 = 2 - i, z_2 = 1 + i,\text { find } \left| \frac{z_1 + z_2 + 1}{z_1 - z_2 + i} \right|\]
If \[x + iy = \frac{a + ib}{a - ib}\] prove that x2 + y2 = 1.
Find the real values of θ for which the complex number \[\frac{1 + i cos\theta}{1 - 2i cos\theta}\] is purely real.
Find the smallest positive integer value of m for which \[\frac{(1 + i )^n}{(1 - i )^{n - 2}}\] is a real number.
If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
If z1, z2, z3 are complex numbers such that \[\left| z_1 \right| = \left| z_2 \right| = \left| z_3 \right| = \left| \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} \right| = 1\] then find the value of \[\left| z_1 + z_2 + z_3 \right|\] .
Express the following complex in the form r(cos θ + i sin θ):
1 − sin α + i cos α
Express the following complex in the form r(cos θ + i sin θ):
\[\frac{1 - i}{\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}}\]
If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of \[\left| z \right|\] .
If \[\left| z - 5i \right| = \left| z + 5i \right|\] , then find the locus of z.
Write the value of \[\sqrt{- 25} \times \sqrt{- 9}\].
Write the value of \[\arg\left( z \right) + \arg\left( \bar{z} \right)\].
If \[\left| z \right| = 2 \text { and } \arg\left( z \right) = \frac{\pi}{4}\],find z.
The polar form of (i25)3 is
The principal value of the amplitude of (1 + i) is
If z is a non-zero complex number, then \[\left| \frac{\left| z \right|^2}{zz} \right|\] is equal to
\[\text { If } z = \frac{1}{(1 - i)(2 + 3i)}, \text { than } \left| z \right| =\]
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
If \[z = \frac{1}{1 - cos\theta - i sin\theta}\] then Re (z) =
The amplitude of \[\frac{1 + i\sqrt{3}}{\sqrt{3} + i}\] is
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
The value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}} - 1\] is
The value of \[(1 + i )^4 + (1 - i )^4\] is
Find a and b if abi = 3a − b + 12i
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
`((1 + "i")/(1 - "i"))^2`
Show that `(-1 + sqrt(3)"i")^3` is a real number
Evaluate the following : `1/"i"^58`
Find the value of `(i^(592) + i^(590) + i^(588) + i^(586) + i^(584))/(i^(582) + i^(580) + i^(578) + i^(576) + i^(574))`
