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
Find the multiplicative inverse of the complex number.
–i
Find the value of i49 + i68 + i89 + i110
Simplify the following and express in the form a + ib:
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
Write the conjugates of the following complex number:
5i
Write the conjugates of the following complex number:
`sqrt(5) - "i"`
Find the value of i49 + i68 + i89 + i110
Find the value of i + i2 + i3 + i4
Simplify:
`(i^592 + i^590 + i^588 + i^586 + i^584)/(i^582 + i^580 + i^578 + i^576 + i^574)`
Show that `((sqrt(7) + "i"sqrt(3))/(sqrt(7) - "i"sqrt(3)) + (sqrt(7) - "i"sqrt(3))/(sqrt(7) + "i"sqrt(3)))` is real
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:
(2 + 3i)(1 − 4i)
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:
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:
`(5 + 7"i")/(4 + 3"i") + (5 + 7"i")/(4 - 3"i")`
Answer the following:
Find the value of x3 + 2x2 − 3x + 21, if x = 1 + 2i
Answer the following:
If x + iy = `("a" + "ib")/("a" - "ib")`, prove that x2 + y2 = 1
If z1 = 5 + 3i and z2 = 2 - 4i, then z1 + z2 = ______.
Evaluate: (1 + i)6 + (1 – i)3
State true or false for the following:
If three complex numbers z1, z2 and z3 are in A.P., then they lie on a circle in the complex plane.
The area of the triangle on the complex plane formed by the complex numbers z, –iz and z + iz is ______.
Number of solutions of the equation z2 + |z|2 = 0 is ______.
If |z + 4| ≤ 3, then the greatest and least values of |z + 1| are ______ and ______.
The value of `(z + 3)(barz + 3)` is equivalent to ______.
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 ______.
If a + ib = c + id, then ______.
If the least and the largest real values of α, for which the equation z + α|z – 1| + 2i = 0 `("z" ∈ "C" and "i" = sqrt(-1))` has a solution, are p and q respectively; then 4(p2 + q2) is equal to ______.
`((1 + cosθ + isinθ)/(1 + cosθ - isinθ))^n` = ______.
If a complex number z satisfies the equation `z + sqrt(2)|z + 1| + i` = 0, then |z| is equal 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)`
Show that `(-1 + sqrt3 i)^3` is a real number.
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)`
i2 + i3 + ... + i4000 =
If z = 2 + i, then (z − 1) `(barz − 5) + (barz − 1)` (z − 5) is equal to ______.
