Advertisements
Advertisements
प्रश्न
For any two complex numbers z1 and z2, such that |z1| = |z2| = 1 and z1 z2 ≠ -1, then show that `(z_1 + z_2)/(1 + z_1 z_2)` is real number
उत्तर
Given |z1| = |z2| = 1 and z1z2 ≠ 1
|z1|2 = 1
|z2|2 = 1
`"z"_1 bar("z")_1` = 1
Similarly `"z"_2 bar("z")_2` = 1
z1 = `1/bar(z)_1`
z2 = `1/bar(z)_2`
`(z_1 + z_2)/(1 + z_1z_2) = (1/bar(z)_1 + 1/bar(z)_2)/(1 + 1/bar(z)_1 1/bar(z)_2)`
= `((bar(z)_2 + bar(z)_1)/(bar(z)_1 + bar(z)_2))/((bar(z)_1 bar(z)_2 + 1)/(bar(z)_1 bar(z)_2))`
= `(bar(z)_1 + bar(z)_2)/(1 + bar(z)_1 bar(z)_2)`
= `(bar(z_1 + z_2))/(bar(1 + z_1 + z_2))`
= `bar(((z_1 + z_2)/(1 + z_1 + z_2))`
∴ `(z_1 + z_2)/(1 + z_1 z_2) = bar(((z_1 + z_2)/(1 + z_1 + z_2))`
∴ `(z_1 + z_2)/(1 + z_1 z_2)` is real.
Since z = `bar(z)` [z is real]
Since z = `bar(z)`, it is a real number.
APPEARS IN
संबंधित प्रश्न
Find the modulus of the following complex numbers
(1 – i)10
Which one of the points 10 – 8i, 11 + 6i is closest to 1 + i
If |z| = 3, show that 7 ≤ |z + 6 – 8i| ≤ 13
If |z| = 2 show that the 8 ≤ |z + 6 + 8i| ≤ 12
If z1, z2 and z3 are three complex numbers such that |z1| = 1, |z2| = 2, |z3| = 3 and |z1 + z2 + z3| = 1, show that |9z1z2 + 4z1z3 + z2z3| = 6
If the area of the triangle formed by the vertices z, iz, and z + iz is 50 square units, find the value of |z|
Find the square roots of 4 + 3i
Find the square roots of – 5 – 12i
Choose the correct alternative:
If = `((sqrt(3) + "i")^3 (3"i" + 4)^2)/(8 + 6"i")^2`, then |z| is equal to
Choose the correct alternative:
If z is a non zero complex number, such that 2iz2 = `bar(z)` then |z| is
Choose the correct alternative:
If |z – 2 + i| ≤ 2, then the greatest value of |z| is
Choose the correct alternative:
If (1 + i)(1 + 2i)(1 + 3i) ……. (l + ni) = x + iy, then 2.5.10 …… (1 + n2) is
Choose the correct alternative:
If α and β are the roots of x² + x + 1 = 0, then α2020 + β2020 is
The square roots of -18i are ______
If α and β are roots of the equation x2 + 5|x| – 6 = 0, then the value of |tan–1α – tan–1 β| is ______.
