Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Let \[z = x + iy\]
Then,
\[z_2 = \frac{z_1 - 1}{z_1 + 1}\]
\[ = \frac{x + iy - 1}{x + iy + 1}\]
\[ = \frac{\left( x - 1 \right) + iy}{\left( x + 1 \right) + iy} \times \frac{\left( x + 1 \right) - iy}{\left( x + 1 \right) - iy}\]
\[ = \frac{x^2 + x - ixy - x - 1 + iy + ixy + iy - i^2 y^2}{\left( x + 1 \right)^2 - i^2 y^2}\]
\[ = \frac{x^2 + y^2 - 1 + 2iy}{x^2 + 1 + 2x + y^2} [ \because i^2 = - 1]\]
Now,
\[Re\left( z_2 \right) = \frac{x^2 + y^2 - 1}{x^2 + y^2 + 1 + 2x}\]
\[ = 0 [ \because \left| z_1 \right| = 1 \Rightarrow x^2 + y^2 = 1]\]
Thus, the real parts of z2 is zero.
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: i9 + i19
Express the given complex number in the form a + ib: (1 – i) – (–1 + i6)
Evaluate the following:
(ii) i528
Evaluate the following:
\[( i^{77} + i^{70} + i^{87} + i^{414} )^3\]
Evaluate the following:
\[i^{49} + i^{68} + i^{89} + i^{110}\]
Find the value of the following expression:
i49 + i68 + i89 + i110
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:
\[(1 + 2i )^{- 3}\]
Find the real value of x and y, if
\[(3x - 2iy)(2 + i )^2 = 10(1 + i)\]
Find the real value of x and y, if
\[(1 + i)(x + iy) = 2 - 5i\]
Find the multiplicative inverse of the following complex number:
\[(1 + i\sqrt{3} )^2\]
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Im `(1/(z_1overlinez_1))`
If \[\frac{\left( 1 + i \right)^2}{2 - i} = x + iy\] 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:
\[x^4 - 4 x^3 + 4 x^2 + 8x + 44,\text { when } x = 3 + 2i\]
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
Evaluate the following:
\[2 x^4 + 5 x^3 + 7 x^2 - x + 41, \text { when } x = - 2 - \sqrt{3}i\]
If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].
Solve the equation \[\left| z \right| = z + 1 + 2i\].
If n is any positive integer, write the value of \[\frac{i^{4n + 1} - i^{4n - 1}}{2}\].
Write the value of \[\sqrt{- 25} \times \sqrt{- 9}\].
If\[z = \cos\frac{\pi}{4} + i \sin\frac{\pi}{6}\], then
If a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]
If (x + iy)1/3 = a + ib, then \[\frac{x}{a} + \frac{y}{b} =\]
The value of (i5 + i6 + i7 + i8 + i9) / (1 + i) is
The value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}} - 1\] is
If z is a complex number, then
Find a and b if a + 2b + 2ai = 4 + 6i
Find a and b if abi = 3a − b + 12i
Find a and b if `1/("a" + "ib")` = 3 – 2i
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)−1
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
`((2 + "i"))/((3 - "i")(1 + 2"i"))`
Find the value of `(3 + 2/i) (i^6 - i^7) (1 + i^11)`.
Evaluate the following : i35
Evaluate the following : i–888
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)`.
If `((1 - i)/(1 + i))^100` = a + ib, then find (a, b).
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
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 |
