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
If α and β are different complex numbers with |β| = 1, then find `|(beta - alpha)/(1-baralphabeta)|`
If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that (a2 + b2) (c2 + d2) (e2 + f2) (g2 + h2) = A2 + B2.
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: 2x3 – 11x2 + 44x + 27, if x = `25/(3 - 4"i")`
Simplify the following and express in the form a + ib:
(1 + 3i)2 (3 + i)
Simplify the following and express in the form a + ib:
`(sqrt(5) + sqrt(3)"i")/(sqrt(5) - sqrt(3)"i")`
Find the value of: x3 – 3x2 + 19x – 20, if x = 1 – 4i
Find the value of i49 + i68 + i89 + i110
Simplify:
`(i^592 + i^590 + i^588 + i^586 + i^584)/(i^582 + i^580 + i^578 + i^576 + i^574)`
Show that 1 + i10 + i100 − i1000 = 0
Find the value of x and y which satisfy the following equation (x, y ∈ R).
`((x + "i"y))/(2 + 3"i") + (2 + "i")/(2 - 3"i") = 9/13(1 + "i")`
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 :
Answer the following:
Simplify the following and express in the form a + ib:
`3 + sqrt(-64)`
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:
Evaluate: i131 + i49
The value of (2 + i)3 × (2 – i)3 is ______.
Evaluate: (1 + i)6 + (1 – i)3
If z1, z2, z3 are complex numbers such that `|z_1| = |z_2| = |z_3| = |1/z_1 + 1/z_2 + 1/z_3|` = 1, then find the value of |z1 + z2 + z3|.
The value of `(- sqrt(-1))^(4"n" - 3)`, where n ∈ N, is ______.
What is the locus of z, if amplitude of z – 2 – 3i is `pi/4`?
If the complex number z = x + iy satisfies the condition |z + 1| = 1, then z lies on ______.
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 ______.
Evaluate `sum_(n = 1)^13 (i^n + i^(n + 1))`, where n ∈ N.
If (1 + i)z = `(1 - i)barz`, then show that z = `-ibarz`.
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.
Find the complex number satisfying the equation `z + sqrt(2) |(z + 1)| + i` = 0.
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.
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).
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 ______.
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)`
Find the value of `sqrt(-3) xx sqrt(-6)`
Show that `(-1 + sqrt3i)^3` is a real number.
