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:
`[(1/3 + i 7/3) + (4 + i 1/3)] -(-4/3 + i)`
Evaluate the following:
\[i^{30} + i^{40} + i^{60}\]
Show that 1 + i10 + i20 + i30 is a real number.
Find the value of the following expression:
1+ i2 + i4 + i6 + i8 + ... + i20
Express the following complex number in the standard form a + i b:
\[\frac{1 - i}{1 + i}\]
Express the following complex number in the standard form a + i b:
\[\frac{(2 + i )^3}{2 + 3i}\]
Express the following complex number in the standard form a + i b:
\[\frac{(1 - i )^3}{1 - i^3}\]
Express the following complex number in the standard form a + i b:
\[\left( \frac{1}{1 - 4i} - \frac{2}{1 + i} \right)\left( \frac{3 - 4i}{5 + i} \right)\]
Find the real value of x and y, if
\[(3x - 2iy)(2 + i )^2 = 10(1 + i)\]
If \[x + iy = \frac{a + ib}{a - ib}\] prove that x2 + y2 = 1.
Evaluate the following:
\[2 x^4 + 5 x^3 + 7 x^2 - x + 41, \text { when } x = - 2 - \sqrt{3}i\]
If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
Write (i25)3 in polar form.
Express the following complex in the form r(cos θ + i sin θ):
tan α − i
Express the following complex in the form r(cos θ + i sin θ):
1 − sin α + i cos α
Write the value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\] .
Find z, if \[\left| z \right| = 4 \text { and }\arg(z) = \frac{5\pi}{6} .\]
If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of \[x^2 + y^2\].
Write the value of \[\sqrt{- 25} \times \sqrt{- 9}\].
Write the value of \[\arg\left( z \right) + \arg\left( \bar{z} \right)\].
Write the argument of \[\left( 1 + i\sqrt{3} \right)\left( 1 + i \right)\left( \cos\theta + i\sin\theta \right)\].
Disclaimer: There is a misprinting in the question. It should be \[\left( 1 + i\sqrt{3} \right)\] instead of \[\left( 1 + \sqrt{3} \right)\].
The value of \[(1 + i)(1 + i^2 )(1 + i^3 )(1 + i^4 )\] is.
If\[z = \cos\frac{\pi}{4} + i \sin\frac{\pi}{6}\], then
If i2 = −1, then the sum i + i2 + i3 +... upto 1000 terms is equal to
If \[z = \frac{- 2}{1 + i\sqrt{3}}\],then the value of arg (z) is
\[(\sqrt{- 2})(\sqrt{- 3})\] is equal to
If \[z = \frac{1 + 7i}{(2 - i )^2}\] , then
The amplitude of \[\frac{1}{i}\] is equal to
The value of (i5 + i6 + i7 + i8 + i9) / (1 + i) is
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
Find a and b if abi = 3a − b + 12i
Find a and b if `1/("a" + "ib")` = 3 – 2i
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
(1 + i)−3
If `((1 + "i"sqrt3)/(1 - "i"sqrt3))^"n"` is an integer, then n is ______.
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
Show that `(-1+ sqrt(3)i)^3` is a real number.
Show that `(-1+sqrt3i)^3` is a real number.
