Advertisements
Advertisements
Question
If x + iy = `sqrt(("a" + "ib")/("c" + "id")`, prove that (x2 + y2)2 = `("a"^2 + "b"^2)/("c"^2 + "d"^2)`
Advertisements
Solution
x + iy = `sqrt(("a" + "ib")/("c" + "id")`
∴ (x + iy)2 = `("a" + "ib")/("c" + "id")`
∴ x2 + 2xyi + y2i2 = `("a" + "ib")/("c" + "id") xx ("c" - "id")/("c" - "id")`
∴ x2 + 2xyi – y2 = `("ac" - "adi" + "bci" - "bdi"^2)/("c"^2 - "d"^2"i"^2)` ...[∵ i2 = –1]
∴ (x2 – y2) + 2xyi = `("ac" - "adi" + "bci" + "bd")/("c"^2 + "d"^2)`
∴ (x2 – y2) + 2xyi =`(("ac" + "bd") + ("bc" - "ad")"i")/("c"^2 + "d"^2)`
∴ (x2 – y2) + 2xyi = `(("ac" + "bd")/("c"^2 + "d"^2)) + (("bc" - "ad")/("c"^2 + "d"^2))"i"`
Equating the real and imaginary parts separately, we get,
x2 – y2 = `("ac" + "bd")/("c"^2 + "d"^2)` and 2xy = `("bc" - "ad")/("c"^2 + "d"^2)`
∴ (x2 + y2)2 = (x2 – y2)2 + 4x2y2
= (x2 – y2)2 + (2xy)2
= `(("ac" + "bd")/("c"^2 + "d"^2))^2 + (("bc" - "ad")/("c"^2 + "d"^2))^2`
= `(("ac" + "bd")^2 + ("bc" - "ad")^2)/("c"^2 + "d"^2)^2`
= `("a"^2"c"^2 + 2"abcd" + "b"^2"d"^2 + "b"^2"c"^2 - 2"abcd" + "a"^2"d"^2)/("c"^2 + "d"^2)^2`
= `(("a"^2"c"^2 + "b"^2"c"^2) + ("a"^2"d"^2 + "b"^2"d"^2))/("c"^2 + "d"^2)^2`
= `(("a"^2 + "b"^2)"c"^2 + ("a"^2 + "b"^2)"d"^2)/("c"^2 + "d"^2)^2`
= `(("a"^2 + "b"^2)("c"^2 + "d"^2))/("c"^2 + "d"^2)^2`
∴ (x2 + y2)2 = `("a"^2 + "b"^2)/("c"^2 + "d"^2)`
APPEARS IN
RELATED QUESTIONS
Reduce `(1/(1-4i) - 2/(1+i))((3-4i)/(5+i))` to the standard form.
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 + 3i)2 (3 + i)
Write the conjugates of the following complex number:
3 – i
Find the value of i49 + i68 + i89 + i110
Is (1 + i14 + i18 + i22) a real number? Justify your answer
Find the value of `("i"^6 + "i"^7 + "i"^8 + "i"^9)/("i"^2 + "i"^3)`
If x + iy = (a + ib)3, show that `x/"a" + y/"b"` = 4(a2 − b2)
If (a + ib) = `(1 + "i")/(1 - "i")`, then prove that (a2 + b2) = 1
If (x + iy)3 = y + vi then show that `(y/x + "v"/y)` = 4(x2 – y2)
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:
(2 + 3i)(1 − 4i)
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:
`(sqrt(5) + sqrt(3)"i")/(sqrt(5) - sqrt(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:
Solve the following equations for x, y ∈ R:
(x + iy) (5 + 6i) = 2 + 3i
Answer the following:
Evaluate: i131 + i49
Answer the following:
Simplify: `("i"^65 + 1/"i"^145)`
Answer the following:
Simplify: `("i"^238 + "i"^236 + "i"^234 + "i"^232 + "i"^230)/("i"^228 + "i"^226 + "i"^224 + "i"^222 + "i"^220)`
The argument of the complex number `(4 + 9i)/(13 + 5i)` is ______
If z1 = 5 + 3i and z2 = 2 - 4i, then z1 + z2 = ______.
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 ______.
The value of `(- sqrt(-1))^(4"n" - 3)`, where n ∈ N, is ______.
State true or false for the following:
The complex number cosθ + isinθ can be zero for some θ.
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.
If |z1| = |z2| = ... = |zn| = 1, then show that |z1 + z2 + z3 + ... + zn| = `|1/z_1 + 1/z_2 + 1/z_3 + ... + 1/z_n|`.
Find the complex number satisfying the equation `z + sqrt(2) |(z + 1)| + i` = 0.
The value of `sqrt(-25) xx sqrt(-9)` is ______.
The sum of the series i + i2 + i3 + ... upto 1000 terms is ______.
State True or False for the following:
Multiplication of a non-zero complex number by –i rotates the point about origin through a right angle in the anti-clockwise direction.
Let |z| = |z – 3| = |z – 4i|, then the value |2z| is ______.
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.
`(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 + sqrt3i)^3` is a real number.
