Advertisements
Advertisements
Question
If \[z = a + ib\] lies in third quadrant, then \[\frac{\bar{z}}{z}\] also lies in third quadrant if
Options
\[a > b > 0\]
\[a < b < 0\]
\[b < a < 0\]
\[b > a > 0\]
Advertisements
Solution
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
RELATED QUESTIONS
Express the given complex number in the form a + ib: `(1/5 + i 2/5) - (4 + i 5/2)`
If a + ib = `(x + i)^2/(2x^2 + 1)` prove that a2 + b2 = `(x^2 + 1)^2/(2x + 1)^2`
Find the value of the following expression:
\[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\]
Express the following complex number in the standard form a + i b:
\[\frac{3 + 2i}{- 2 + i}\]
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 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 \[\left( \frac{1 + i}{1 - i} \right)^3 - \left( \frac{1 - i}{1 + i} \right)^3 = x + iy\] find (x, y).
If \[a = \cos\theta + i\sin\theta\], find the value of \[\frac{1 + a}{1 - a}\].
Evaluate the following:
\[2 x^3 + 2 x^2 - 7x + 72, \text { when } x = \frac{3 - 5i}{2}\]
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
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
Express the following complex in the form r(cos θ + i sin θ):
1 − sin α + i cos α
Write −1 + i \[\sqrt{3}\] in polar form .
Write the argument of −i.
Write the value of \[\arg\left( z \right) + \arg\left( \bar{z} \right)\].
If `(3+2i sintheta)/(1-2 i sin theta)`is a real number and 0 < θ < 2π, then θ =
The polar form of (i25)3 is
If \[z = \frac{- 2}{1 + i\sqrt{3}}\],then the value of arg (z) is
If a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]
If θ is the amplitude of \[\frac{a + ib}{a - ib}\] , than tan θ =
The value of \[(1 + i )^4 + (1 - i )^4\] is
Simplify : `4sqrt(-4) + 5sqrt(-9) - 3sqrt(-16)`
Find a and b if (a – b) + (a + b)i = a + 5i
Find a and b if (a+b) (2 + i) = b + 1 + (10 + 2a)i
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
`("i"(4 + 3"i"))/((1 - "i"))`
Evaluate the following : i116
Evaluate the following : i30 + i40 + i50 + i60
If `((1 + "i"sqrt3)/(1 - "i"sqrt3))^"n"` is an integer, then n is ______.
If z1 = 3 – 2i and z2 = –1 + 3i, then Im(z1z2) = ______.
If `((1 - i)/(1 + i))^100` = a + ib, then find (a, b).
Show that `(-1+ sqrt(3)i)^3` is a real number.
