Advertisements
Advertisements
प्रश्न
For any two complex numbers z1, z2 and any real numbers a, b, |az1 – bz2|2 + |bz1 + az2|2 = ______.
उत्तर
For any two complex numbers z1, z2 and any real numbers a, b, |az1 – bz2|2 + |bz1 + az2|2 = `(a^2 + b^2)(|z_1|^2 + |z2|^2)`.
Explanation:
|az1 – bz2|2 + |bz1 + az2|2
= `|az_1|^2 + |bz_2|^2 – 2 "Re"(az
_1 . b barz_2) + |bz_1|^2 + |az_2|^2 + 2 "Re" (az_1 . b barz_2)`
= `|az_1|^2 + |bz_2|^2 + |bz_1|^2 + |az_2|^2`
= `(a^2 + b^2)(|z_1|^2 + |z_2|^2)`
APPEARS IN
संबंधित प्रश्न
If α and β are different complex numbers with |β| = 1, then find `|(beta - alpha)/(1-baralphabeta)|`
Find the value of: x3 – x2 + x + 46, if x = 2 + 3i
Simplify the following and express in the form a + ib:
(2i3)2
Simplify the following and express in the form a + ib:
(2 + 3i)(1 – 4i)
Write the conjugates of the following complex number:
5i
Find the value of i + i2 + i3 + i4
Prove that `(1 + "i")^4 xx (1 + 1/"i")^4` = 16
Find the value of x and y which satisfy the following equation (x, y∈R).
(x + 2y) + (2x − 3y)i + 4i = 5
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:
(1 + 3i)2(3 + i)
Answer the following:
Find the value of x3 + 2x2 − 3x + 21, if x = 1 + 2i
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)`
The real value of ‘a’ for which 3i3 – 2ai2 + (1 – a)i + 5 is real is ______.
If (2 + i) (2 + 2i) (2 + 3i) ... (2 + ni) = x + iy, then 5.8.13 ... (4 + n2) = ______.
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.
Solve the equation |z| = z + 1 + 2i.
Find the complex number satisfying the equation `z + sqrt(2) |(z + 1)| + i` = 0.
The sum of the series i + i2 + i3 + ... upto 1000 terms is ______.
If z1 and z2 are complex numbers such that z1 + z2 is a real number, then z2 = ______.
Find `|(1 + i) ((2 + i))/((3 + i))|`.
The value of `(z + 3)(barz + 3)` is equivalent to ______.
A real value of x satisfies the equation `((3 - 4ix)/(3 + 4ix))` = α − iβ (α, β ∈ R) if α2 + β2 = ______.
If z is a complex number, then ______.
`((1 + cosθ + isinθ)/(1 + cosθ - isinθ))^n` = ______.
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 + sqrt3 i)^3` is a real number.
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 `sqrt(-3) xx sqrt(-6)`
