Advertisements
Advertisements
प्रश्न
If \[z = a + ib\] lies in third quadrant, then \[\frac{\bar{z}}{z}\] also lies in third quadrant if
पर्याय
\[a > b > 0\]
\[a < b < 0\]
\[b < a < 0\]
\[b > a > 0\]
Advertisements
उत्तर
Since, \[z = a + ib\] lies in third quadrant. \[\Rightarrow a < 0 \text { and } b < 0 . . . . (1)\]
Now,
\[\frac{\bar{z}}{z} = \frac{\bar{{a + ib}}}{a + ib}\]
\[ = \frac{a - ib}{a + ib}\]
\[ = \frac{a - ib}{a + ib} \times \frac{a - ib}{a - ib}\]
\[ = \frac{a^2 + i^2 b^2 - 2abi}{a^2 - i^2 b^2}\]
\[ = \frac{a^2 - b^2 - 2abi}{a^2 + b^2}\]
Since,
\[\frac{\bar{z}}{z}\] also lies in third quadrant.
\[\Rightarrow a^2 - b^2 < 0\]
\[ \Rightarrow (a - b)(a + b) < 0\]
\[ \Rightarrow a - b > 0 \text { and }a + b < 0\]
\[ \Rightarrow a > b . . . . (2)\]
From (1) and (2),
\[b < a < 0\]
Hence, the correct option is (c).
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: 3(7 + i7) + i(7 + i7)
Show that 1 + i10 + i20 + i30 is a real number.
Find the value of the following expression:
i49 + i68 + i89 + i110
Find the value of the following expression:
i5 + i10 + i15
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:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
Find the real value of x and y, if
\[(x + iy)(2 - 3i) = 4 + i\]
Find the multiplicative inverse of the following complex number:
\[(1 + i\sqrt{3} )^2\]
If \[\frac{\left( 1 + i \right)^2}{2 - i} = x + iy\] find x + y.
Evaluate the following:
\[2 x^4 + 5 x^3 + 7 x^2 - x + 41, \text { when } x = - 2 - \sqrt{3}i\]
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 \[\frac{z - 1}{z + 1}\] is purely imaginary number (\[z \neq - 1\]), find the value of \[\left| z \right|\].
If \[\left| z + 1 \right| = z + 2\left( 1 + i \right)\],find z.
Find the number of solutions of \[z^2 + \left| z \right|^2 = 0\].
Write (i25)3 in polar form.
Express the following complex in the form r(cos θ + i sin θ):
1 + i tan 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 \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of \[x^2 + y^2\].
If `(3+2i sintheta)/(1-2 i sin theta)`is a real number and 0 < θ < 2π, then θ =
If\[z = \cos\frac{\pi}{4} + i \sin\frac{\pi}{6}\], then
The polar form of (i25)3 is
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
The least positive integer n such that \[\left( \frac{2i}{1 + i} \right)^n\] is a positive integer, is.
\[\text { If } z = \frac{1}{(2 + 3i )^2}, \text { than } \left| z \right| =\]
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
The amplitude of \[\frac{1}{i}\] is equal to
The argument of \[\frac{1 - i}{1 + i}\] is
A real value of x satisfies the equation \[\frac{3 - 4ix}{3 + 4ix} = a - ib (a, b \in \mathbb{R}), if a^2 + b^2 =\]
Simplify : `sqrt(-16) + 3sqrt(-25) + sqrt(-36) - sqrt(-625)`
Express the following in the form of a + ib, a, b ∈ R i = `sqrt(−1)`. State the values of a and b:
`(2 + sqrt(-3))/(4 + sqrt(-3))`
Express the following in the form of a + ib, a, b ∈ R i = `sqrt(−1)`. State the values of a and b:
`(- sqrt(5) + 2sqrt(-4)) + (1 -sqrt(-9)) + (2 + 3"i")(2 - 3"i")`
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)`
Match the statements of column A and B.
| Column A | Column B |
| (a) The value of 1 + i2 + i4 + i6 + ... i20 is | (i) purely imaginary complex number |
| (b) The value of `i^(-1097)` is | (ii) purely real complex number |
| (c) Conjugate of 1 + i lies in | (iii) second quadrant |
| (d) `(1 + 2i)/(1 - i)` lies in | (iv) Fourth quadrant |
| (e) If a, b, c ∈ R and b2 – 4ac < 0, then the roots of the equation ax2 + bx + c = 0 are non real (complex) and |
(v) may not occur in conjugate pairs |
| (f) If a, b, c ∈ R and b2 – 4ac > 0, and b2 – 4ac is a perfect square, then the roots of the equation ax2 + bx + c = 0 |
(vi) may occur in conjugate pairs |
Find the value of `(i^(592) + i^(590) + i^(588) + i^(586) + i^(584))/(i^(582) + i^(580) + i^(578) + i^(576) + i^(574))`
