Advertisements
Advertisements
प्रश्न
Solve the equation \[\left| z \right| = z + 1 + 2i\].
Advertisements
उत्तर
Let \[z = x + iy\]
Then,
\[\left| z \right| = \sqrt{x^2 + y^2}\]
\[\therefore \left| z \right| = z + 1 + 2i\]
\[ \Rightarrow \sqrt{x^2 + y^2} = \left( x + iy \right) + 1 + 2i\]
\[ \Rightarrow \sqrt{x^2 + y^2} = \left( x + 1 \right) + i\left( y + 2 \right)\]
\[ \Rightarrow \sqrt{x^2 + y^2} = \left( x + 1 \right) \text { and } y + 2 = 0\]
\[ \Rightarrow x^2 + y^2 = \left( x + 1 \right)^2 \text { and } y = - 2\]
\[ \Rightarrow x^2 + y^2 = x^2 + 1 + 2x \text { and } y = - 2\]
\[ \Rightarrow y^2 = 2x + 1\text { and } y = - 2\]
\[ \Rightarrow 4 = 2x + 1 \text { and } y = - 2\]
\[ \Rightarrow 2x = 3 \text { and } y = - 2\]
\[ \Rightarrow x = \frac{3}{2} \text { and } y = - 2\]
\[\therefore z = x + iy = \frac{3}{2} - 2i\]
Thus,
\[z = \frac{3}{2} - 2i\]
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:
`[(1/3 + i 7/3) + (4 + i 1/3)] -(-4/3 + i)`
Evaluate: `[i^18 + (1/i)^25]^3`
Let z1 = 2 – i, z2 = –2 + i. Find Re`((z_1z_2)/barz_1)`
Express the following complex number in the standard form a + i b:
\[(1 + i)(1 + 2i)\]
Express the following complex number in the standard form a + i b:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
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
\[(3x - 2iy)(2 + i )^2 = 10(1 + i)\]
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
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Re \[\left( \frac{z_1 z_2}{z_1} \right)\]
Find the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
Find the smallest positive integer value of m for which \[\frac{(1 + i )^n}{(1 - i )^{n - 2}}\] is a real number.
Evaluate the following:
\[x^6 + x^4 + x^2 + 1, \text { when }x = \frac{1 + i}{\sqrt{2}}\]
If \[\frac{z - 1}{z + 1}\] is purely imaginary number (\[z \neq - 1\]), find the value of \[\left| z \right|\].
What is the smallest positive integer n for which \[\left( 1 + i \right)^{2n} = \left( 1 - i \right)^{2n}\] ?
Express the following complex in the form r(cos θ + i sin θ):
tan α − i
If n is any positive integer, write the value of \[\frac{i^{4n + 1} - i^{4n - 1}}{2}\].
Write the value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\] .
Write the argument of −i.
Write the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
Write the value of \[\arg\left( z \right) + \arg\left( \bar{z} \right)\].
If \[\left| z + 4 \right| \leq 3\], then find the greatest and least values of \[\left| z + 1 \right|\].
If `(3+2i sintheta)/(1-2 i sin theta)`is a real number and 0 < θ < 2π, then θ =
If a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]
The argument of \[\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\] is
\[\text { If } z = \frac{1}{(2 + 3i )^2}, \text { than } \left| z \right| =\]
The argument of \[\frac{1 - i}{1 + i}\] is
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
`((2 + "i"))/((3 - "i")(1 + 2"i"))`
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
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
`(4"i"^8 - 3"i"^9 + 3)/(3"i"^11 - 4"i"^10 - 2)`
Evaluate the following : i35
Evaluate the following : i93
Evaluate the following : i116
Show that 1 + i10 + i20 + i30 is a real number
Answer the following:
Show that z = `5/((1 - "i")(2 - "i")(3 - "i"))` is purely imaginary number.
If z1 = 3 – 2i and z2 = –1 + 3i, then Im(z1z2) = ______.
The real value of θ for which the expression `(1 + i cos theta)/(1 - 2i cos theta)` is a real number is ______.
Show that `(-1+ sqrt(3)i)^3` is a real number.
