Advertisements
Advertisements
Question
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
Solution
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
RELATED QUESTIONS
Find the multiplicative inverse of the complex number.
–i
Express the following expression in the form of a + ib.
`((3 + sqrt5)(3 - isqrt5))/((sqrt3 + sqrt2i)-(sqrt3 - isqrt2))`
If `x – iy = sqrt((a-ib)/(c - id))` prove that `(x^2 + y^2) = (a^2 + b^2)/(c^2 + d^2)`
If `((1+i)/(1-i))^m` = 1, then find the least positive integral value of m.
Simplify the following and express in the form a + ib:
`3 + sqrt(-64)`
Simplify the following and express in the form a + ib:
`(sqrt(5) + sqrt(3)"i")/(sqrt(5) - sqrt(3)"i")`
Write the conjugates of the following complex number:
`sqrt(5) - "i"`
Write the conjugates of the following complex number:
`sqrt(2) + sqrt(3)"i"`
Write the conjugates of the following complex number:
cosθ + i sinθ
Find the value of i49 + i68 + i89 + i110
Show that 1 + i10 + i100 − i1000 = 0
Is (1 + i14 + i18 + i22) a real number? Justify your answer
Prove that `(1 + "i")^4 xx (1 + 1/"i")^4` = 16
If `("a" + 3"i")/(2+ "ib")` = 1 − i, show that (5a − 7b) = 0
Show that `((sqrt(7) + "i"sqrt(3))/(sqrt(7) - "i"sqrt(3)) + (sqrt(7) - "i"sqrt(3))/(sqrt(7) + "i"sqrt(3)))` is real
Find the value of x and y which satisfy the following equation (x, y∈R).
(x + 2y) + (2x − 3y)i + 4i = 5
Select the correct answer from the given alternatives:
The value of is `("i"^592 + "i"^590 + "i"^588 + "i"^586 + "i"^584)/("i"^582 + "i"^580 + "i"^578 + "i"^576 + "i"^574)` is equal to:
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:
`5/2"i"(-4 - 3"i")`
Answer the following:
Simplify the following and express in the form a + ib:
`(4 + 3"i")/(1 - "i")`
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:
Solve the following equation for x, y ∈ R:
(4 − 5i)x + (2 + 3i)y = 10 − 7i
What is the principal value of amplitude of 1 – i?
1 + i2 + i4 + i6 + ... + i2n is ______.
If the complex number z = x + iy satisfies the condition |z + 1| = 1, then z lies on ______.
The equation |z + 1 – i| = |z – 1 + i| represents a ______.
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 |z + 4| ≤ 3, then the greatest and least values of |z + 1| are ______ and ______.
State True or False for the following:
The inequality |z – 4| < |z – 2| represents the region given by x > 3.
The value of `(z + 3)(barz + 3)` is equivalent to ______.
Let x, y ∈ R, then x + iy is a non-real complex number if ______.
If a + ib = c + id, then ______.
Find the value of `(i^592 + i^590 + i^588 + i^586 + i^584)/ (i^582 + i^580 + i^578 + i^576 + i^574)`
Simplify the following and express in the form a + ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
Simplify the following and express in the form a + ib.
`(3i^5+2i^7+i^9)/(i^6+2i^8+3i^18)`
Simplify the following and express in the form a+ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18`
