Advertisements
Advertisements
प्रश्न
If θ is the amplitude of \[\frac{a + ib}{a - ib}\] , than tan θ =
पर्याय
\[\frac{2a}{a^2 + b^2}\]
\[\frac{2ab}{a^2 - b^2}\]
\[\frac{a^2 - b^2}{a^2 + b^2}\]
none of these
Advertisements
उत्तर
\[\frac{2ab}{a^2 - b^2}\]
\[z = \frac{a + ib}{a - ib} \times \frac{a + ib}{a + ib}\]
\[ \Rightarrow z = \frac{a^2 + i^2 b^2 + 2abi}{a^2 - i^2 b^2}\]
\[ \Rightarrow z = \frac{a^2 - b^2 + 2abi}{a^2 + b^2}\]
\[ \Rightarrow z = \frac{a^2 - b^2}{a^2 + b^2} + i\frac{2ab}{a^2 + b^2}\]
\[ \Rightarrow \text { Re }\left( z \right) = \frac{a^2 - b^2}{a^2 + b^2}, \text { Im }\left( z \right) = \frac{2ab}{a^2 + b^2}\]
\[\tan \alpha = \left| \frac{Im\left( z \right)}{Re\left( z \right)} \right|\]
\[ = \frac{2ab}{a^2 - b^2}\]
\[\alpha = \tan^{- 1} \left( \frac{2ab}{a^2 - b^2} \right)\]
\[\text { Since, z lies in the first quadrant . Therefore, } \]
\[\arg (z) = \alpha = \tan^{- 1} \left( \frac{2ab}{a^2 - b^2} \right)\]
\[\tan \theta = \frac{2ab}{a^2 - b^2}\]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: (1 – i) – (–1 + i6)
Evaluate the following:
\[\frac{1}{i^{58}}\]
Find the value of the following expression:
i30 + i80 + i120
Express the following complex number in the standard form a + ib:
\[\frac{(2 + i )^3}{2 + 3i}\]
Find the multiplicative inverse of the following complex number:
\[(1 + i\sqrt{3} )^2\]
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 \[\frac{1 + a}{1 - a}\].
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.
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 n is any positive integer, write the value of \[\frac{i^{4n + 1} - i^{4n - 1}}{2}\].
Write 1 − i in polar form.
Write −1 + i \[\sqrt{3}\] in polar form .
Write the argument of −i.
If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of \[x^2 + y^2\].
Write the value of \[\sqrt{- 25} \times \sqrt{- 9}\].
Find the real value of a for which \[3 i^3 - 2a i^2 + (1 - a)i + 5\] is real.
If \[\left| z \right| = 2 \text { and } \arg\left( z \right) = \frac{\pi}{4}\],find z.
Write the argument of \[\left( 1 + i\sqrt{3} \right)\left( 1 + i \right)\left( \cos\theta + i\sin\theta \right)\].
Disclaimer: There is a misprinting in the question. It should be \[\left( 1 + i\sqrt{3} \right)\] instead of \[\left( 1 + \sqrt{3} \right)\].
If `(3+2i sintheta)/(1-2 i sin theta)`is a real number and 0 < θ < 2π, then θ =
If \[z = \frac{1 + 2i}{1 - (1 - i )^2}\], then arg (z) equal
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
The amplitude of \[\frac{1 + i\sqrt{3}}{\sqrt{3} + i}\] is
The value of (i5 + i6 + i7 + i8 + i9) / (1 + i) is
If \[z = a + ib\] lies in third quadrant, then \[\frac{\bar{z}}{z}\] also lies in third quadrant if
If \[f\left( z \right) = \frac{7 - z}{1 - z^2}\] , where \[z = 1 + 2i\] then \[\left| f\left( z \right) \right|\] is
Find a and b if a + 2b + 2ai = 4 + 6i
Find a and b if (a – b) + (a + b)i = a + 5i
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))`
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 |
State True or False for the following:
The order relation is defined on the set of complex numbers.
