Advertisements
Advertisements
प्रश्न
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)`
Advertisements
उत्तर
L.H.S. = `|1 - barz_1z_2|^2 - |z_1 - z_2|^2`
= `(1 - barz_1z_2) (bar(1 - barz_1 z_2)) - (z_1 - z_2) (bar(z_1 - z_2))`
= `(1 - barz_1 z_2) (1 - z_1 barz_2) - (z_1 - z_2)(barz_1 - barz_2)`
= `1 + z_1 barz_1 z_2barz_2 - z_1barz_1 - z_2barz_2`
= `1 + |z-1|^2 * |z_2|^2 - |z_1|^2 - |z_2|^2`
= `(1 - |z_1|^2)(1 - |z_2|^2)`
R.H.S. = `k(1 - |z_1|^2)(1 - |z_2|^2)`
⇒ k = 1
Hence, equating L.H.S. and R.H.S., we get k = 1.
APPEARS IN
संबंधित प्रश्न
Find the value of: x3 – x2 + x + 46, if x = 2 + 3i
Find the value of: x3 – 5x2 + 4x + 8, if x = `10/(3 - "i")`.
Write the conjugates of the following complex number:
3 – i
Write the conjugates of the following complex number:
`-sqrt(-5)`
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)`
Find the value of 1 + i2 + i4 + i6 + i8 + ... + i20
Find the value of x and y which satisfy the following equation (x, y∈R).
`(x+ 1)/(1 + "i") + (y - 1)/(1 - "i")` = i
Select the correct answer from the given alternatives:
If n is an odd positive integer then the value of 1 + (i)2n + (i)4n + (i)6n is :
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:
`(sqrt(5) + sqrt(3)"i")/(sqrt(5) - sqrt(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 x4 + 9x3 + 35x2 − x + 164, if x = −5 + 4i
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")`
Answer the following:
Show that z = `((-1 + sqrt(-3))/2)^3` is a rational number
If z1 = 5 + 3i and z2 = 2 - 4i, then z1 + z2 = ______.
If z1 = 2 – 4i and z2 = 1 + 2i, then `bar"z"_1 + bar"z"_2` = ______.
The value of (2 + i)3 × (2 – i)3 is ______.
Evaluate: (1 + i)6 + (1 – i)3
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 value of `(i^(4n + 1) -i^(4n - 1))/2`?
The area of the triangle on the complex plane formed by the complex numbers z, –iz and z + iz is ______.
Number of solutions of the equation z2 + |z|2 = 0 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 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| = 1(z1 ≠ –1) and z2 = `(z_1 - 1)/(z_1 + 1)`, then show that the real part of z2 is zero.
Multiplicative inverse of 1 + i is ______.
If z1 and z2 are complex numbers such that z1 + z2 is a real number, then z2 = ______.
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 ______.
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 ______.
If α and β are the roots of the equation x2 + 2x + 4 = 0, then `1/α^3 + 1/β^3` is equal to ______.
If `|(6i, -3i, 1),(4, 3i, -1),(20, 3, i)|` = x + iy, then ______.
Simplify the following and express in the form a + ib.
`(3i^5+2i^7+i^9)/(i^6+2i^8+3i^18)`
Evaluate the following:
i35
