Advertisements
Advertisements
प्रश्न
For any two complex numbers z1, z2 and any real numbers a, b, |az1 – bz2|2 + |bz1 + az2|2 = ______.
Advertisements
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
Reduce `(1/(1-4i) - 2/(1+i))((3-4i)/(5+i))` to the standard form.
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:
(2i3)2
Simplify the following and express in the form a + ib:
(2 + 3i)(1 – 4i)
Write the conjugates of the following complex number:
3 + i
Write the conjugates of the following complex number:
3 – i
Write the conjugates of the following complex number:
`-sqrt(-5)`
Write the conjugates of the following complex number:
5i
Find the value of i + i2 + i3 + i4
Simplify:
`(i^592 + i^590 + i^588 + i^586 + i^584)/(i^582 + i^580 + i^578 + i^576 + i^574)`
Show that 1 + i10 + i100 − i1000 = 0
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).
`(x+ 1)/(1 + "i") + (y - 1)/(1 - "i")` = i
Answer the following:
Simplify the following and express in the form a + ib:
`3 + sqrt(-64)`
Answer the following:
Simplify the following and express in the form a + ib:
(2i3)2
Answer the following:
Simplify the following and express in the form a + ib:
(1 + 3i)2(3 + i)
Answer the following:
Evaluate: (1 − i + i2)−15
Answer the following:
Find the value of x3 + 2x2 − 3x + 21, if x = 1 + 2i
Answer the following:
Simplify: `("i"^29 + "i"^39 + "i"^49)/("i"^30 + "i"^40 + "i"^50)`
Answer the following:
Simplify `[1/(1 - 2"i") + 3/(1 + "i")] [(3 + 4"i")/(2 - 4"i")]`
The equation |z + 1 – i| = |z – 1 + i| represents a ______.
Evaluate `sum_(n = 1)^13 (i^n + i^(n + 1))`, where n ∈ N.
If `(1 + i)^2/(2 - i)` = x + iy, then find the value of x + y.
If the real part of `(barz + 2)/(barz - 1)` is 4, then show that the locus of the point representing z in the complex plane is 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 = ______.
If |z + 4| ≤ 3, then the greatest and least values of |z + 1| are ______ and ______.
State True or False for the following:
For any complex number z the minimum value of |z| + |z – 1| is 1.
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).
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 ______.
Let |z| = |z – 3| = |z – 4i|, then the value |2z| is ______.
If `(x + iy)^(1/5)` = a + ib, and u = `x/a - y/b`, 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)`
