Advertisements
Advertisements
Question
If θ is the amplitude of \[\frac{a + ib}{a - ib}\] , than tan θ =
Options
\[\frac{2a}{a^2 + b^2}\]
\[\frac{2ab}{a^2 - b^2}\]
\[\frac{a^2 - b^2}{a^2 + b^2}\]
none of these
Advertisements
Solution
\[\frac{2ab}{a^2 - b^2}\]
\[z = \frac{a + ib}{a - ib} \times \frac{a + ib}{a + ib}\]
\[ \Rightarrow z = \frac{a^2 + i^2 b^2 + 2abi}{a^2 - i^2 b^2}\]
\[ \Rightarrow z = \frac{a^2 - b^2 + 2abi}{a^2 + b^2}\]
\[ \Rightarrow z = \frac{a^2 - b^2}{a^2 + b^2} + i\frac{2ab}{a^2 + b^2}\]
\[ \Rightarrow \text { Re }\left( z \right) = \frac{a^2 - b^2}{a^2 + b^2}, \text { Im }\left( z \right) = \frac{2ab}{a^2 + b^2}\]
\[\tan \alpha = \left| \frac{Im\left( z \right)}{Re\left( z \right)} \right|\]
\[ = \frac{2ab}{a^2 - b^2}\]
\[\alpha = \tan^{- 1} \left( \frac{2ab}{a^2 - b^2} \right)\]
\[\text { Since, z lies in the first quadrant . Therefore, } \]
\[\arg (z) = \alpha = \tan^{- 1} \left( \frac{2ab}{a^2 - b^2} \right)\]
\[\tan \theta = \frac{2ab}{a^2 - b^2}\]
APPEARS IN
RELATED QUESTIONS
Let z1 = 2 – i, z2 = –2 + i. Find Re`((z_1z_2)/barz_1)`
Evaluate the following:
\[\frac{1}{i^{58}}\]
Evaluate the following:
\[i^{30} + i^{40} + i^{60}\]
Find the value of the following expression:
(1 + i)6 + (1 − i)3
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:
\[(1 + i)(1 + 2i)\]
Express the following complex number in the standard form a + i b:
\[\frac{1 - i}{1 + i}\]
Find the real value of x and y, if
\[(x + iy)(2 - 3i) = 4 + i\]
Find the real value of x and y, if `((1+i)x-2i)/(3+i) + ((2-3i)y+i)/(3-i) = i, xy ∈ R, i = sqrt-1`
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:
\[x^6 + x^4 + x^2 + 1, \text { when }x = \frac{1 + i}{\sqrt{2}}\]
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
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 \[\left( 1 + i \right)^{2n} = \left( 1 - i \right)^{2n}\] ?
If n is any positive integer, write the value of \[\frac{i^{4n + 1} - i^{4n - 1}}{2}\].
Write −1 + i \[\sqrt{3}\] in polar form .
Write the argument of −i.
Write the value of \[\sqrt{- 25} \times \sqrt{- 9}\].
If \[\left| z + 4 \right| \leq 3\], then find the greatest and least values of \[\left| z + 1 \right|\].
For any two complex numbers z1 and z2 and any two real numbers a, b, find the value of \[\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2\].
If n ∈ \[\mathbb{N}\] then find the value of \[i^n + i^{n + 1} + i^{n + 2} + i^{n + 3}\] .
If\[z = \cos\frac{\pi}{4} + i \sin\frac{\pi}{6}\], then
If \[x + iy = \frac{3 + 5i}{7 - 6i},\] then y =
The argument of \[\frac{1 - i}{1 + i}\] is
The amplitude of \[\frac{1 + i\sqrt{3}}{\sqrt{3} + i}\] is
If z is a complex number, then
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:
`((1 + "i")/(1 - "i"))^2`
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:
(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:
`(2 + sqrt(-3))/(4 + sqrt(-3))`
Evaluate the following : i35
Evaluate the following : i93
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 `((1 + "i"sqrt3)/(1 - "i"sqrt3))^"n"` is an integer, then n is ______.
Match the statements of Column A and Column B.
| Column A | Column B |
| (a) The polar form of `i + sqrt(3)` is | (i) Perpendicular bisector of segment joining (–2, 0) and (2, 0). |
| (b) The amplitude of `-1 + sqrt(-3)` is | (ii) On or outside the circle having centre at (0, –4) and radius 3. |
| (c) If |z + 2| = |z − 2|, then locus of z is | (iii) `(2pi)/3` |
| (d) If |z + 2i| = |z − 2i|, then locus of z is | (iv) Perpendicular bisector of segment joining (0, –2) and (0, 2). |
| (e) Region represented by |z + 4i| ≥ 3 is | (v) `2(cos pi/6 + i sin pi/6)` |
| (f) Region represented by |z + 4| ≤ 3 is | (vi) On or inside the circle having centre (–4, 0) and radius 3 units. |
| (g) Conjugate of `(1 + 2i)/(1 - i)` lies in | (vii) First quadrant |
| (h) Reciprocal of 1 – i lies in | (viii) Third quadrant |
