Advertisements
Advertisements
प्रश्न
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
उत्तर
`|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
संबंधित प्रश्न
If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that (a2 + b2) (c2 + d2) (e2 + f2) (g2 + h2) = A2 + B2.
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:
`(4 + 3"i")/(1 - "i")`
Simplify the following and express in the form a + ib:
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
Find the value of: x3 – 5x2 + 4x + 8, if x = `10/(3 - "i")`.
Find the value of i49 + i68 + i89 + i110
Is (1 + i14 + i18 + i22) a real number? Justify your answer
If x + iy = `sqrt(("a" + "ib")/("c" + "id")`, prove that (x2 + y2)2 = `("a"^2 + "b"^2)/("c"^2 + "d"^2)`
If (a + ib) = `(1 + "i")/(1 - "i")`, then prove that (a2 + b2) = 1
If (x + iy)3 = y + vi then show that `(y/x + "v"/y)` = 4(x2 – y2)
Find the value of x and y which satisfy the following equation (x, y∈R).
`(x+ 1)/(1 + "i") + (y - 1)/(1 - "i")` = i
Answer the following:
Simplify the following and express in the form a + ib:
`5/2"i"(-4 - 3"i")`
Answer the following:
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)`
Answer the following:
Find the value of x3 + 2x2 − 3x + 21, if x = 1 + 2i
Answer the following:
Find the real numbers x and y such that `x/(1 + 2"i") + y/(3 + 2"i") = (5 + 6"i")/(-1 + 8"i")`
Show that `(1/sqrt(2) + "i"/sqrt(2))^10 + (1/sqrt(2) - "i"/sqrt(2))^10` = 0
If z1 = 5 + 3i and z2 = 2 - 4i, then z1 + z2 = ______.
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 reciprocal of `3 + sqrt(7)i`.
What is the principal value of amplitude of 1 – i?
What is the locus of z, if amplitude of z – 2 – 3i is `pi/4`?
If the complex number z = x + iy satisfies the condition |z + 1| = 1, then z lies on ______.
For a positive integer n, find the value of `(1 - i)^n (1 - 1/i)^"n"`
If `((1 + i)/(1 - i))^3 - ((1 - i)/(1 + i))^3` = x + iy, then find (x, y).
For any two complex numbers z1, z2 and any real numbers a, b, |az1 – bz2|2 + |bz1 + az2|2 = ______.
The number `(1 - i)^3/(1 - i^2)` is equal to ______.
The sum of the series i + i2 + i3 + ... upto 1000 terms is ______.
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).
`((1 + cosθ + isinθ)/(1 + cosθ - isinθ))^n` = ______.
If a complex number z satisfies the equation `z + sqrt(2)|z + 1| + i` = 0, then |z| is equal to ______.
The complex number z = x + iy which satisfy the equation `|(z - 5i)/(z + 5i)|` = 1, lie on ______.
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)`
Evaluate the following:
i35
