Advertisements
Advertisements
Question
If z1, z2, z3 are complex numbers such that `|z_1| = |z_2| = |z_3| = |1/z_1 + 1/z_2 + 1/z_3|` = 1, then find the value of |z1 + z2 + z3|.
Advertisements
Solution
`|z_1| = |z_2| = |z_3|` = 1
⇒ `|z_1|^2 = |z_2|^2 = |z_3|^2` = 1
⇒ `z_1 barz_1 = z_2 barz_2 = z_3 barz_3` = 1
⇒ `barz_1 = 1/barz_1, barz_2 = 1/barz_2, barz_3 = 1/z_3`
Given that `|1/z_1 + 1/z_2 + 1/z_3|` = 1
⇒ `|barz_1 + barz_2 + barz_3|` = 1, i.e., `|bar(z_1 + z_2 + z_3)|` = 1
⇒ |z1 + z2 + z3| = 1
APPEARS IN
RELATED QUESTIONS
Find the number of non-zero integral solutions of the equation `|1-i|^x = 2^x`.
Find the value of i49 + i68 + i89 + i110
Write the conjugates of the following complex number:
3 + i
Write the conjugates of the following complex number:
`-sqrt(5) - sqrt(7)"i"`
Write the conjugates of the following complex number:
`sqrt(5) - "i"`
Find the value of i49 + i68 + i89 + i110
Evaluate: `("i"^37 + 1/"i"^67)`
Find the value of `("i"^6 + "i"^7 + "i"^8 + "i"^9)/("i"^2 + "i"^3)`
If x + iy = (a + ib)3, show that `x/"a" + y/"b"` = 4(a2 − b2)
If (a + ib) = `(1 + "i")/(1 - "i")`, then prove that (a2 + b2) = 1
Find the value of x and y which satisfy the following equation (x, y∈R).
If x(1 + 3i) + y(2 − i) − 5 + i3 = 0, find x + y
Find the value of x and y which satisfy the following equation (x, y∈R).
If x + 2i + 15i6y = 7x + i3 (y + 4), find x + y
Answer the following:
Simplify the following and express in the form a + ib:
(2 + 3i)(1 − 4i)
Answer the following:
Simplify the following and express in the form a + ib:
`5/2"i"(-4 - 3"i")`
Answer the following:
show that `((1 + "i")/sqrt(2))^8 + ((1 - "i")/sqrt(2))^8` = 2
Answer the following:
Show that z = `((-1 + sqrt(-3))/2)^3` is a rational number
Answer the following:
Show that `(1 - 2"i")/(3 - 4"i") + (1 + 2"i")/(3 + 4"i")` is real
If |z1| = 1, |z2| = 2, |z3| = 3 and |9z1z2 + 4z1z3 + z2z3| = 12, then the value of |z1 + z2 + z3| is
State true or false for the following:
Multiplication of a non-zero complex number by i rotates it through a right angle in the anti-clockwise direction.
What is the locus of z, if amplitude of z – 2 – 3i is `pi/4`?
1 + i2 + i4 + i6 + ... + i2n is ______.
If `(1 + i)^2/(2 - i)` = x + iy, then find the value of x + y.
If z = x + iy, then show that `z barz + 2(z + barz) + b` = 0, where b ∈ R, represents a circle.
If |z + 1| = z + 2(1 + i), then find z.
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 = ______.
State True or False for the following:
The locus represented by |z – 1| = |z – i| is a line perpendicular to the join of (1, 0) and (0, 1).
Where does z lie, if `|(z - 5i)/(z + 5i)|` = 1.
A real value of x satisfies the equation `((3 - 4ix)/(3 + 4ix))` = α − iβ (α, β ∈ R) if α2 + β2 = ______.
Let |z| = |z – 3| = |z – 4i|, then the value |2z| is ______.
`((1 + cosθ + isinθ)/(1 + cosθ - isinθ))^n` = ______.
The smallest positive integer n for which `((1 + i)/(1 - i))^n` = –1 is ______.
Let z be a complex number such that `|(z - i)/(z + 2i)|` = 1 and |z| = `5/2`. Then the value of |z + 3i| is ______.
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 `sqrt(-3) xx sqrt(-6)`
i2 + i3 + ... + i4000 =
