Advertisements
Advertisements
Question
If α and β are different complex numbers with |β| = 1, then find `|(beta - alpha)/(1-baralphabeta)|`
Solution
`|(beta - alpha)/(1 - baralpha beta)|^2`
= `((beta - alpha)/(1 - baralpha beta))bar(((beta - alpha)/(1 - baralpha beta))`
= `(beta - alpha)/(1 - baralpha beta) xx (barbeta - baralpha)/(1 - baralpha beta)`
= `(beta barbeta - baralphabeta - alpha barbeta + alpha baralpha)/(1 - alpha barbeta - baralphabeta +alphabaralpha.betabarbeta)`
= `(|beta|^2 - baralphabeta - alphabarbeta + |alpha|^2)/(1 - alphabarbeta - baralphabeta + |alpha|^2 . |beta|^2`)`
Given |β| = 1,
= `(1 + |alpha|^2 - baralphabeta - alphabarbeta)/(1 + |alpha|^2 - baralphabeta - alphabarbeta)`
= 1
∴ `|(beta - alpha)/(1 - baralphabeta)| = 1` or `|(beta - alpha)/(1 - baralphabeta)| = 1`
APPEARS IN
RELATED QUESTIONS
Show that 1 + i10 + i20 + i30 is a real number.
Find the value of: 2x3 – 11x2 + 44x + 27, if x = `25/(3 - 4"i")`
Simplify the following and express in the form a + ib:
(2 + 3i)(1 – 4i)
Find the value of : x3 + 2x2 – 3x + 21, if x = 1 + 2i
Write the conjugates of the following complex number:
3 – i
Write the conjugates of the following complex number:
`sqrt(2) + sqrt(3)"i"`
If a = `(-1 + sqrt(3)"i")/2`, b = `(-1 - sqrt(3)"i")/2` then show that a2 = b and b2 = a
Select the correct answer from the given alternatives:
`sqrt(-3) sqrt(-6)` is equal to
Answer the following:
Simplify the following and express in the form a + ib:
(2 + 3i)(1 − 4i)
Answer the following:
Solve the following equation for x, y ∈ R:
`(x + "i"y)/(2 + 3"i")` = 7 – i
Answer the following:
Solve the following equations for x, y ∈ R:
(x + iy) (5 + 6i) = 2 + 3i
Answer the following:
Evaluate: (1 − i + i2)−15
Answer the following:
Find the value of x4 + 9x3 + 35x2 − x + 164, if x = −5 + 4i
Answer the following:
Show that `(1/sqrt(2) + "i"/sqrt(2))^10 + (1/sqrt(2) - "i"/sqrt(2))^10` = 0
Answer the following:
show that `((1 + "i")/sqrt(2))^8 + ((1 - "i")/sqrt(2))^8` = 2
Answer the following:
Show that `(1 - 2"i")/(3 - 4"i") + (1 + 2"i")/(3 + 4"i")` is real
If z1 = 2 – 4i and z2 = 1 + 2i, then `bar"z"_1 + bar"z"_2` = ______.
State true or false for the following:
The argument of the complex number z = `(1 + i sqrt(3))(1 + i)(cos theta + i sin theta)` is `(7pi)/12 + theta`.
What is the principal value of amplitude of 1 – i?
1 + i2 + i4 + i6 + ... + i2n is ______.
The equation |z + 1 – i| = |z – 1 + i| represents a ______.
If `(1 + i)^2/(2 - i)` = x + iy, then find the value of x + y.
If `(z - 1)/(z + 1)` is purely imaginary number (z ≠ – 1), then find the value of |z|.
If |z1| = 1(z1 ≠ –1) and z2 = `(z_1 - 1)/(z_1 + 1)`, then show that the real part of z2 is zero.
If |z1| = |z2| = ... = |zn| = 1, then show that |z1 + z2 + z3 + ... + zn| = `|1/z_1 + 1/z_2 + 1/z_3 + ... + 1/z_n|`.
If z1 and z2 are complex numbers such that z1 + z2 is a real number, then z2 = ______.
Where does z lie, if `|(z - 5i)/(z + 5i)|` = 1.
`((1 + cosθ + isinθ)/(1 + cosθ - isinθ))^n` = ______.
If `|(6i, -3i, 1),(4, 3i, -1),(20, 3, i)|` = x + iy, then ______.
Let `(-2 - 1/3i)^2 = (x + iy)/9 (i = sqrt(-1))`, where x and y are real numbers, then x – y equals to ______.
Find the value of `(i^592+i^590+i^588+i^586+i^584)/(i^582+i^580+i^578+i^576+i^574)`
Find the value of `(i^592 + i^590 + i^588 + i^586 + i^584)/ (i^582 + i^580 + i^578 + i^576 + i^574)`
Simplify the following and express in the form a + ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
Simplify the following and express in the form a+ib:
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
