Advertisements
Advertisements
प्रश्न
Answer the following:
If x + iy = `("a" + "ib")/("a" - "ib")`, prove that x2 + y2 = 1
Advertisements
उत्तर
x + iy = `("a" + "ib")/("a" - "ib") = (("a" + "ib")("a" + "ib"))/(("a" - "ib")("a" + "ib"))`
= `("a"^2 + "i"^2"b"^2 + 2"abi")/("a"^2 - "i"^2"b"^2)`
= `(("a"^2 - "b"^2) + 2"abi")/("a"^2 + "b"^2)` ...[∵ i2 = – 1]
∴ x + iy = `("a"^2 - "b"^2)/("a"^2 + "b"^2) + (2"ab")/("a"^2 + "b"^2)"i"`
Equating real and imaginary parts, we get
x = `("a"^2 - "b"^2)/("a"^2 + "b"^2)` and y = `(2"ab")/("a"^2 + "b"^2)`
∴ x2 + y2 = `(("a"^2 - "b"^2)^2)/("a"^2 + "b"^2)^2 + (4"a"^2"b"^2)/("a"^2 + "b"^2)^2`
= `("a"^4 + "b"^4 - 2"a"^2 "b"^2 + 4"a"^2"b"^2)/("a"^2 + "b"^2)^2`
= `(("a"^2 + "b"^2)^2)/("a"^2 + "b"^2)^2`
∴ x2 + y2 = 1
APPEARS IN
संबंधित प्रश्न
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)`
Show that 1 + i10 + i20 + i30 is a real number.
Find the value of i49 + i68 + i89 + i110
Find the value of i + i2 + i3 + i4
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:
3 – i
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 x + iy = `sqrt(("a" + "ib")/("c" + "id")`, prove that (x2 + y2)2 = `("a"^2 + "b"^2)/("c"^2 + "d"^2)`
If (a + ib) = `(1 + "i")/(1 - "i")`, then prove that (a2 + b2) = 1
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 :
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:
Select the correct answer from the given alternatives:
`sqrt(-3) sqrt(-6)` is equal to
Answer the following:
Simplify the following and express in the form a + ib:
(1 + 3i)2(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:
Evaluate: i131 + i49
If z1 = 2 – 4i and z2 = 1 + 2i, then `bar"z"_1 + bar"z"_2` = ______.
Find the value of 2x4 + 5x3 + 7x2 – x + 41, when x = `-2 - sqrt(3)"i"`.
What is the value of `(i^(4n + 1) -i^(4n - 1))/2`?
What is the locus of z, if amplitude of z – 2 – 3i is `pi/4`?
If z = x + iy, then show that `z barz + 2(z + barz) + b` = 0, where b ∈ R, represents a circle.
If |z + 1| = z + 2(1 + i), then find z.
Find the complex number satisfying the equation `z + sqrt(2) |(z + 1)| + i` = 0.
The number `(1 - i)^3/(1 - i^2)` is equal to ______.
The sum of the series i + i2 + i3 + ... upto 1000 terms is ______.
If z1 and z2 are complex numbers such that z1 + z2 is a real number, then z2 = ______.
Find `|(1 + i) ((2 + i))/((3 + i))|`.
The real value of α for which the expression `(1 - i sin alpha)/(1 + 2i sin alpha)` is purely real is ______.
The value of `(z + 3)(barz + 3)` is equivalent to ______.
If `((1 + i)/(1 - i))^x` = 1, then ______.
A real value of x satisfies the equation `((3 - 4ix)/(3 + 4ix))` = α − iβ (α, β ∈ R) if α2 + β2 = ______.
If z is a complex number, then ______.
If `(x + iy)^(1/5)` = a + ib, and u = `x/a - y/b`, then ______.
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)`
Show that `(-1 + sqrt3 i)^3` is a real number.
If z = 2 + i, then (z − 1) `(barz − 5) + (barz − 1)` (z − 5) is equal to ______.
