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
Reduce `(1/(1-4i) - 2/(1+i))((3-4i)/(5+i))` to the standard form.
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.
Show that 1 + i10 + i20 + i30 is a real number.
Find the value of: x3 – x2 + x + 46, if x = 2 + 3i
Write the conjugates of the following complex number:
3 + i
Find the value of i49 + i68 + i89 + i110
Find the value of x and y which satisfy the following equation (x, y∈R).
(x + 2y) + (2x − 3y)i + 4i = 5
Find the value of x and y which satisfy the following equation (x, y∈R).
If x(1 + 3i) + y(2 − i) − 5 + i3 = 0, find x + y
Answer the following:
Simplify the following and express in the form a + ib:
`(1 + 2/"i")(3 + 4/"i")(5 + "i")^-1`
Answer the following:
Find the value of x4 + 9x3 + 35x2 − x + 164, if x = −5 + 4i
Answer the following:
If x + iy = `("a" + "ib")/("a" - "ib")`, prove that x2 + y2 = 1
Answer the following:
Show that `(1 - 2"i")/(3 - 4"i") + (1 + 2"i")/(3 + 4"i")` is real
Answer the following:
Simplify `[1/(1 - 2"i") + 3/(1 + "i")] [(3 + 4"i")/(2 - 4"i")]`
If z ≠ 1 and `"z"^2/("z - 1")` is real, then the point represented by the complex number z lies ______.
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)`
The real value of ‘a’ for which 3i3 – 2ai2 + (1 – a)i + 5 is real is ______.
If (2 + i) (2 + 2i) (2 + 3i) ... (2 + ni) = x + iy, then 5.8.13 ... (4 + n2) = ______.
State true or false for the following:
Multiplication of a non-zero complex number by i rotates it through a right angle in the anti-clockwise direction.
State true or false for the following:
The points representing the complex number z for which |z + 1| < |z − 1| lies in the interior of a circle.
State true or false for the following:
If n is a positive integer, then the value of in + (i)n+1 + (i)n+2 + (i)n+3 is 0.
What is the locus of z, if amplitude of z – 2 – 3i is `pi/4`?
Evaluate `sum_(n = 1)^13 (i^n + i^(n + 1))`, where n ∈ N.
Solve the equation |z| = z + 1 + 2i.
If |z1| = 1(z1 ≠ –1) and z2 = `(z_1 - 1)/(z_1 + 1)`, then show that the real part of z2 is zero.
The value of `(z + 3)(barz + 3)` is equivalent to ______.
Which of the following is correct for any two complex numbers z1 and z2?
The point represented by the complex number 2 – i is rotated about origin through an angle `pi/2` in the clockwise direction, the new position of point is ______.
Let x, y ∈ R, then x + iy is a non-real complex number if ______.
If a + ib = c + id, then ______.
If `(3 + i)(z + barz) - (2 + i)(z - barz) + 14i` = 0, then `barzz` is equal 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.
`(3"i"^5 + 2"i"^7 + "i"^9)/("i"^6 + 2"i"^8 + 3"i"^18)`
Simplify the following and express in the form a + ib.
`(3i^5 + 2i^7 + i^9) / (i^6 + 2i^8 + 3i^18)`
Find the value of `sqrt(-3) xx sqrt(-6)`
Simplify the following and express in the form a+ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
