Advertisements
Advertisements
Question
If |z1| = |z2| = ... = |zn| = 1, then show that |z1 + z2 + z3 + ... + zn| = `|1/z_1 + 1/z_2 + 1/z_3 + ... + 1/z_n|`.
Solution
We have |z1| = |z2| = ... = |zn| = 1
⇒ |z1|2 = |z2|2 = ... = |zn|2 = 1 ......(i)
⇒ `z_1 barz_1 = z_2 barz_2 = ... = z_n barz_n` = 1 .....`[because zbarz = |z|^2]`
⇒ z1 = `1/barz_1, z_2 = 1/barz_2 = ... = z_n = 1/barz_n`
L.H.S. |z1 + z2 + z3 + ... + zn|
= `|(z_1barz_1)/barz_1 + (z_2barz_2)/barz_2 + (z_3barz_3)/barz_3 + ... + (z_nbarz_n)/barz_n|`
= `||z_1|^2/barz_1 + (|z_2|^2)/barz_2 + (|z_3|^2)/barz_3 + ... + (|z_n|^2)/barz_n|` ......`[zbarz = |z|^2]`
= `|1/barz_1 + 1/barz_2 + 1/barz_3 + ... + 1/barz_n|` ......[Using (i)]
= `|bar(1/z_1 + 1/z_2 + 1/z_3 + ... + 1/z_n)|` .....`[because barz_1 + barz_2 = bar(z_1 + z_2)]`
= `|1/z_1 + 1/z_2 + 1/z_3 + ... + 1/z_n|` ....`[because |z| = |barz|]`
L.H.S. = R.H.S.
Hence proved.
APPEARS IN
RELATED QUESTIONS
Find the multiplicative inverse of the complex number.
–i
If α and β are different complex numbers with |β| = 1, then find `|(beta - alpha)/(1-baralphabeta)|`
Find the number of non-zero integral solutions of the equation `|1-i|^x = 2^x`.
If `((1+i)/(1-i))^m` = 1, then find the least positive integral value of m.
Find the value of: x3 – x2 + x + 46, if x = 2 + 3i
Simplify the following and express in the form a + ib:
`5/2"i"(- 4 - 3 "i")`
Simplify the following and express in the form a + ib:
(1 + 3i)2 (3 + i)
Simplify the following and express in the form a + ib:
`(3"i"^5 + 2"i"^7 + "i"^9)/("i"^6 + 2"i"^8 + 3"i"^18)`
Find the value of i + i2 + i3 + i4
Show that 1 + i10 + i100 − i1000 = 0
Answer the following:
Simplify the following and express in the form a + ib:
(1 + 3i)2(3 + i)
Answer the following:
Solve the following equations for x, y ∈ R:
(x + iy) (5 + 6i) = 2 + 3i
Solve the following equation for x, y ∈ R:
2x + i9y (2 + i) = xi7 + 10i16
If z1 = 2 – 4i and z2 = 1 + 2i, then `bar"z"_1 + bar"z"_2` = ______.
Find the value of k if for the complex numbers z1 and z2, `|1 - barz_1z_2|^2 - |z_1 - z_2|^2 = k(1 - |z_1|^2)(1 - |"z"_2|^2)`
State true or false for the following:
The complex number cosθ + isinθ can be zero for some θ.
State true or false for the following:
If three complex numbers z1, z2 and z3 are in A.P., then they lie on a circle in the complex plane.
What is the value of `(i^(4n + 1) -i^(4n - 1))/2`?
What is the reciprocal of `3 + sqrt(7)i`.
The area of the triangle on the complex plane formed by the complex numbers z, –iz and z + iz is ______.
The value of `sqrt(-25) xx sqrt(-9)` is ______.
The number `(1 - i)^3/(1 - i^2)` is equal to ______.
If z1 and z2 are complex numbers such that z1 + z2 is a real number, then z2 = ______.
The real value of α for which the expression `(1 - i sin alpha)/(1 + 2i sin alpha)` is purely real is ______.
If a + ib = c + id, then ______.
If z1, z2, z3 are complex numbers such that |z1| = |z2| = |z3| = `|1/z_1 + 1/z_2 + 1/z_3|` = 1, then |z1 + z2 + z3| is ______.
If `(3 + i)(z + barz) - (2 + i)(z - barz) + 14i` = 0, then `barzz` is equal to ______.
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)`
Show that `(-1 + sqrt3i)^3` is a real number.
