Advertisements
Advertisements
प्रश्न
The complex number z which satisfies the condition \[\left| \frac{i + z}{i - z} \right| = 1\] lies on
पर्याय
circle x2 + y2 = 1
the x−axis
the y−axis
the line x + y = 1
Advertisements
उत्तर
\[\left| \frac{i + z}{i - z} \right| = 1\]
\[ \Rightarrow \left| \frac{i + z}{i - z} \right|^2 = 1^2 \]
\[ \Rightarrow \left( \frac{i + z}{i - z} \right) \bar{\left( \frac{i + z}{i - z} \right)} = 1\]
\[ \Rightarrow \left( \frac{i + z}{i - z} \right)\left( \frac{- i + \bar{z}}{- i - \bar{z}} \right) = 1\]
\[ \Rightarrow \left( \frac{- i^2 - zi + \bar{z}i + z \bar{z}}{- i^2 + zi - \bar{z}i + z \bar{z}} \right) = 1\]
\[ \Rightarrow - i^2 - zi + \bar{z}i + z \bar{z} = - i^2 + zi - \bar{z}i + z \bar{z}\]
\[ \Rightarrow - zi + \bar{z}i = zi - \bar{z}i\]
\[ \Rightarrow \bar{z}i + \bar{z}i = zi + zi\]
\[ \Rightarrow 2 \bar{z}i = 2zi\]
\[ \Rightarrow \bar{z} = z\]
\[ \Rightarrow \text { z is purely real }\]
Hence, the correct option is (b).
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: (1 – i) – (–1 + i6)
Let z1 = 2 – i, z2 = –2 + i. Find `"Im"(1/(z_1barz_1))`
Evaluate the following:
\[( i^{77} + i^{70} + i^{87} + i^{414} )^3\]
Show that 1 + i10 + i20 + i30 is a real number.
Find the value of the following expression:
i5 + i10 + i15
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:
\[(1 + i)(1 + 2i)\]
Express the following complex number in the standard form a + i b:
\[\frac{(1 + i)(1 + \sqrt{3}i)}{1 - i}\] .
Express the following complex number in the standard form a + i b:
\[\frac{2 + 3i}{4 + 5i}\]
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:
\[\frac{5 + \sqrt{2}i}{1 - 2\sqrt{i}}\]
Find the real value of x and y, if
\[(x + iy)(2 - 3i) = 4 + i\]
Find the real value of x and y, if
\[(3x - 2iy)(2 + i )^2 = 10(1 + i)\]
Find the multiplicative inverse of the following complex number:
\[(1 + i\sqrt{3} )^2\]
Find the real values of θ for which the complex number \[\frac{1 + i cos\theta}{1 - 2i cos\theta}\] is purely real.
If z1 is a complex number other than −1 such that \[\left| z_1 \right| = 1\] and \[z_2 = \frac{z_1 - 1}{z_1 + 1}\] then show that the real parts of z2 is zero.
If \[\left| z + 1 \right| = z + 2\left( 1 + i \right)\],find z.
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 \[\left| z_1 + z_2 + z_3 \right|\] .
Write the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
Find z, if \[\left| z \right| = 4 \text { and }\arg(z) = \frac{5\pi}{6} .\]
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 \[\frac{1 + a}{1 - a} =\]
If \[z = \frac{1}{1 - cos\theta - i sin\theta}\] then Re (z) =
If \[z = \frac{1 + 7i}{(2 - i )^2}\] , then
The argument of \[\frac{1 - i}{1 + i}\] is
The amplitude of \[\frac{1 + i\sqrt{3}}{\sqrt{3} + i}\] is
The value of (i5 + i6 + i7 + i8 + i9) / (1 + i) is
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
A real value of x satisfies the equation \[\frac{3 - 4ix}{3 + 4ix} = a - ib (a, b \in \mathbb{R}), if a^2 + b^2 =\]
Find a and b if a + 2b + 2ai = 4 + 6i
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:
`(3 + 2"i")/(2 - 5"i") + (3 -2"i")/(2 + 5"i")`
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
(2 + 3i)(2 – 3i)
Evaluate the following : i35
Evaluate the following : i30 + i40 + i50 + i60
Answer the following:
Show that z = `5/((1 - "i")(2 - "i")(3 - "i"))` is purely imaginary number.
If z1 and z2 both satisfy `z + barz = 2|z - 1|` arg`(z_1 - z_2) = pi/4`, then find `"Im" (z_1 + z_2)`.
