Advertisements
Advertisements
प्रश्न
If `(z - 1)/(z + 1)` is purely imaginary number (z ≠ – 1), then find the value of |z|.
Advertisements
उत्तर
Given that `(z - 1)/(z + 1)` is purely imaginary number.
Let z = x + yi
∴ `(x + yi - 1)/(x + yi + 1) = ((x - 1) + iy)/((x + 1) + iy)`
= `((x - 1) + iy)/((x + 1) + iy) xx ((x + 1) - iy)/((x + 1) - iy)`
⇒ `((x - 1)(x + 1) - iy(x - 1) + (x + 1)iy - i^2y^2)/((x + 1)^2 - i^2y^2)`
⇒ `(x^2 - 1 + iy(x + 1 - x + 1) + y^2)/(x^2 + 1 + 2x + y^2) = (x^2 + y^2 - 1 + 2yi)/(x^2 + y^2 + 2x + 1)`
⇒ `(x^2 + y^2 - 1)/(x^2 + y^2 + 2x + 1) + (2y)/(x^2 + y^2 + 2x + 1)"i"`
Since, the number is purely imaginary, then real part = 0
∴ `(x^2 + y^2 - 1)/(x^2 + y^2 + 2x + 1)` = 0
⇒ x2 + y2 – 1 = 0
⇒ x2 + y2 = 1
⇒ `sqrt(x^2 + y^2)` = 1
∴ |z| = 1
APPEARS IN
संबंधित प्रश्न
If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that (a2 + b2) (c2 + d2) (e2 + f2) (g2 + h2) = A2 + B2.
Find the value of: x3 – x2 + x + 46, if x = 2 + 3i
Simplify the following and express in the form a + ib:
(2 + 3i)(1 – 4i)
Simplify the following and express in the form a + ib:
(1 + 3i)2 (3 + i)
Simplify the following and express in the form a + ib:
`(1 + 2/i)(3 + 4/i)(5 + i)^-1`
Write the conjugates of the following complex number:
5i
If x + iy = `sqrt(("a" + "ib")/("c" + "id")`, prove that (x2 + y2)2 = `("a"^2 + "b"^2)/("c"^2 + "d"^2)`
Find the value of x and y which satisfy the following equation (x, y∈R).
`(x+ 1)/(1 + "i") + (y - 1)/(1 - "i")` = i
Find the value of x and y which satisfy the following equation (x, y ∈ R).
`((x + "i"y))/(2 + 3"i") + (2 + "i")/(2 - 3"i") = 9/13(1 + "i")`
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:
`5/2"i"(-4 - 3"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:
Evaluate: (1 − i + i2)−15
Answer the following:
Show that `(1 - 2"i")/(3 - 4"i") + (1 + 2"i")/(3 + 4"i")` is real
Answer the following:
Simplify: `("i"^238 + "i"^236 + "i"^234 + "i"^232 + "i"^230)/("i"^228 + "i"^226 + "i"^224 + "i"^222 + "i"^220)`
Answer the following:
Simplify `[1/(1 - 2"i") + 3/(1 + "i")] [(3 + 4"i")/(2 - 4"i")]`
If |z1| = 1, |z2| = 2, |z3| = 3 and |9z1z2 + 4z1z3 + z2z3| = 12, then the value of |z1 + z2 + z3| is
The argument of the complex number `(4 + 9i)/(13 + 5i)` is ______
Locate the points for which 3 < |z| < 4.
If (2 + i) (2 + 2i) (2 + 3i) ... (2 + ni) = x + iy, then 5.8.13 ... (4 + n2) = ______.
State true or false for the following:
The complex number cosθ + isinθ can be zero for some θ.
What is the smallest positive integer n, for which (1 + i)2n = (1 – i)2n?
What is the locus of z, if amplitude of z – 2 – 3i is `pi/4`?
For a positive integer n, find the value of `(1 - i)^n (1 - 1/i)^"n"`
If |z1| = 1(z1 ≠ –1) and z2 = `(z_1 - 1)/(z_1 + 1)`, then show that the real part of z2 is zero.
For any two complex numbers z1, z2 and any real numbers a, b, |az1 – bz2|2 + |bz1 + az2|2 = ______.
The value of `sqrt(-25) xx sqrt(-9)` is ______.
Find `|(1 + i) ((2 + i))/((3 + i))|`.
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 z1, z2, z3 are complex numbers such that |z1| = |z2| = |z3| = `|1/z_1 + 1/z_2 + 1/z_3|` = 1, then |z1 + z2 + z3| is ______.
If α and β are the roots of the equation x2 + 2x + 4 = 0, then `1/α^3 + 1/β^3` is equal to ______.
Let `(-2 - 1/3i)^2 = (x + iy)/9 (i = sqrt(-1))`, where x and y are real numbers, then x – y equals to ______.
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.
`(3"i"^5 + 2"i"^7 + "i"^9)/("i"^6 + 2"i"^8 + 3"i"^18)`
i2 + i3 + ... + i4000 =
