Advertisements
Advertisements
Question
For any two complex numbers z1, z2 and any real numbers a, b, |az1 – bz2|2 + |bz1 + az2|2 = ______.
Advertisements
Solution
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
RELATED QUESTIONS
If `((1+i)/(1-i))^m` = 1, then find the least positive integral value of m.
Find the value of i + i2 + i3 + i4
Simplify the following and express in the form a + ib:
`(sqrt(5) + sqrt(3)"i")/(sqrt(5) - sqrt(3)"i")`
Simplify the following and express in the form a + ib:
`(5 + 7"i")/(4 + 3"i") + (5 + 7"i")/(4 - 3"i")`
Find the value of : x3 + 2x2 – 3x + 21, if x = 1 + 2i
Find the value of: x3 – 3x2 + 19x – 20, if x = 1 – 4i
Write the conjugates of the following complex number:
`sqrt(5) - "i"`
Prove that `(1 + "i")^4 xx (1 + 1/"i")^4` = 16
Find the value of `("i"^6 + "i"^7 + "i"^8 + "i"^9)/("i"^2 + "i"^3)`
If x + iy = `sqrt(("a" + "ib")/("c" + "id")`, prove that (x2 + y2)2 = `("a"^2 + "b"^2)/("c"^2 + "d"^2)`
If (x + iy)3 = y + vi then show that `(y/x + "v"/y)` = 4(x2 – y2)
Answer the following:
Simplify the following and express in the form a + ib:
`3 + sqrt(-64)`
Answer the following:
Evaluate: (1 − i + i2)−15
Answer the following:
Find the value of x4 + 9x3 + 35x2 − x + 164, if x = −5 + 4i
Answer the following:
Simplify: `("i"^65 + 1/"i"^145)`
Locate the points for which 3 < |z| < 4.
Find the value of 2x4 + 5x3 + 7x2 – x + 41, when x = `-2 - sqrt(3)"i"`.
What is the reciprocal of `3 + sqrt(7)i`.
The equation |z + 1 – i| = |z – 1 + i| represents a ______.
Number of solutions of the equation z2 + |z|2 = 0 is ______.
If `((1 + i)/(1 - i))^3 - ((1 - i)/(1 + i))^3` = x + iy, then find (x, y).
If (1 + i)z = `(1 - i)barz`, then show that z = `-ibarz`.
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.
Solve the equation |z| = z + 1 + 2i.
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|`.
The sum of the series i + i2 + i3 + ... upto 1000 terms is ______.
If `((1 + i)/(1 - i))^x` = 1, then ______.
The complex number z which satisfies the condition `|(i + z)/(i - z)|` = 1 lies on ______.
If `(3 + i)(z + barz) - (2 + i)(z - barz) + 14i` = 0, then `barzz` is equal to ______.
A complex number z is moving on `arg((z - 1)/(z + 1)) = π/2`. If the probability that `arg((z^3 -1)/(z^3 + 1)) = π/2` is `m/n`, where m, n ∈ prime, then (m + n) is equal to ______.
Let z be a complex number such that `|(z - i)/(z + 2i)|` = 1 and |z| = `5/2`. Then the value of |z + 3i| is ______.
Let `(-2 - 1/3i)^2 = (x + iy)/9 (i = sqrt(-1))`, where x and y are real numbers, then x – y equals to ______.
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.
Evaluate the following:
i35
Show that `(-1 + sqrt3i)^3` is a real number.
