Advertisements
Advertisements
Question
If (a + ib) = `(1 + "i")/(1 - "i")`, then prove that (a2 + b2) = 1
Advertisements
Solution
(a + ib) = `(1 + "i")/(1 - "i") = (1 + "i")^2/((1 - "i")(1 + "i"))`
∴ a + bi = `(1 + 2"i" + "i"^2)/(1 - "i"^2)`
= `(1 + 2"i" - 1)/(1 - (- 1))` ...[∵ i2 = – 1]
= `(2"i")/2`
= i
∴ a + bi = 0 + i
Equating real and imaginary parts, we get
a = 0 and b = 1
∴ a2 + b2 = 02 + 12 = 1
APPEARS IN
RELATED QUESTIONS
Find the multiplicative inverse of the complex number:
4 – 3i
Simplify the following and express in the form a + ib:
`(5 + 7"i")/(4 + 3"i") + (5 + 7"i")/(4 - 3"i")`
Write the conjugates of the following complex number:
3 + i
Write the conjugates of the following complex number:
`-sqrt(5) - sqrt(7)"i"`
Write the conjugates of the following complex number:
`sqrt(2) + sqrt(3)"i"`
Show that 1 + i10 + i100 − i1000 = 0
Is (1 + i14 + i18 + i22) a real number? Justify your answer
If a = `(-1 + sqrt(3)"i")/2`, b = `(-1 - sqrt(3)"i")/2` then show that a2 = b and b2 = a
If `("a" + 3"i")/(2+ "ib")` = 1 − i, show that (5a − 7b) = 0
Find the value of x and y which satisfy the following equation (x, y∈R).
`(x+ 1)/(1 + "i") + (y - 1)/(1 - "i")` = i
Answer the following:
Simplify the following and express in the form a + ib:
(2i3)2
Solve the following equation for x, y ∈ R:
(4 − 5i)x + (2 + 3i)y = 10 − 7i
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")`
Show that `(1/sqrt(2) + "i"/sqrt(2))^10 + (1/sqrt(2) - "i"/sqrt(2))^10` = 0
Answer the following:
show that `((1 + "i")/sqrt(2))^8 + ((1 - "i")/sqrt(2))^8` = 2
Answer the following:
Show that z = `((-1 + sqrt(-3))/2)^3` is a rational number
If |z1| = 1, |z2| = 2, |z3| = 3 and |9z1z2 + 4z1z3 + z2z3| = 12, then the value of |z1 + z2 + z3| is
If `(x + iy)^(1/3)` = a + ib, where x, y, a, b ∈ R, show that `x/a - y/b` = –2(a2 + b2)
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)`
What is the reciprocal of `3 + sqrt(7)i`.
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.
If |z + 1| = z + 2(1 + i), then find z.
State True or False for the following:
The locus represented by |z – 1| = |z – i| is a line perpendicular to the join of (1, 0) and (0, 1).
Where does z lie, if `|(z - 5i)/(z + 5i)|` = 1.
A real value of x satisfies the equation `((3 - 4ix)/(3 + 4ix))` = α − iβ (α, β ∈ R) if α2 + β2 = ______.
If a + ib = c + id, then ______.
The complex number z which satisfies the condition `|(i + z)/(i - z)|` = 1 lies on ______.
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 ______.
`((1 + cosθ + isinθ)/(1 + cosθ - isinθ))^n` = ______.
The smallest positive integer n for which `((1 + i)/(1 - i))^n` = –1 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)`
Simplify the following and express in the form a+ib:
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
