Advertisements
Advertisements
प्रश्न
If α and β are different complex numbers with |β| = 1, then find `|(beta - alpha)/(1-baralphabeta)|`
उत्तर
`|(beta - alpha)/(1 - baralpha beta)|^2`
= `((beta - alpha)/(1 - baralpha beta))bar(((beta - alpha)/(1 - baralpha beta))`
= `(beta - alpha)/(1 - baralpha beta) xx (barbeta - baralpha)/(1 - baralpha beta)`
= `(beta barbeta - baralphabeta - alpha barbeta + alpha baralpha)/(1 - alpha barbeta - baralphabeta +alphabaralpha.betabarbeta)`
= `(|beta|^2 - baralphabeta - alphabarbeta + |alpha|^2)/(1 - alphabarbeta - baralphabeta + |alpha|^2 . |beta|^2`)`
Given |β| = 1,
= `(1 + |alpha|^2 - baralphabeta - alphabarbeta)/(1 + |alpha|^2 - baralphabeta - alphabarbeta)`
= 1
∴ `|(beta - alpha)/(1 - baralphabeta)| = 1` or `|(beta - alpha)/(1 - baralphabeta)| = 1`
APPEARS IN
संबंधित प्रश्न
Find the multiplicative inverse of the complex number.
–i
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.
Simplify the following and express in the form a + ib:
(2i3)2
Simplify the following and express in the form a + ib:
(2 + 3i)(1 – 4i)
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)`
Simplify the following and express in the form a + ib:
`(5 + 7"i")/(4 + 3"i") + (5 + 7"i")/(4 - 3"i")`
Find the value of 1 + i2 + i4 + i6 + i8 + ... + i20
If x + iy = `sqrt(("a" + "ib")/("c" + "id")`, prove that (x2 + y2)2 = `("a"^2 + "b"^2)/("c"^2 + "d"^2)`
Find the value of x and y which satisfy the following equation (x, y∈R).
(x + 2y) + (2x − 3y)i + 4i = 5
Answer the following:
Simplify the following and express in the form a + ib:
`3 + sqrt(-64)`
Answer the following:
Solve the following equations for x, y ∈ R:
(x + iy) (5 + 6i) = 2 + 3i
Answer the following:
Find the value of x4 + 9x3 + 35x2 − x + 164, if x = −5 + 4i
Answer the following:
Show that z = `((-1 + sqrt(-3))/2)^3` is a rational number
The argument of the complex number `(4 + 9i)/(13 + 5i)` is ______
Evaluate: (1 + i)6 + (1 – i)3
Locate the points for which 3 < |z| < 4.
1 + i2 + i4 + i6 + ... + i2n is ______.
For a positive integer n, find the value of `(1 - i)^n (1 - 1/i)^"n"`
If `((1 + i)/(1 - i))^3 - ((1 - i)/(1 + i))^3` = x + iy, then find (x, y).
If |z + 1| = z + 2(1 + i), then find z.
If |z1| = 1(z1 ≠ –1) and z2 = `(z_1 - 1)/(z_1 + 1)`, then show that the real part of z2 is zero.
Find the complex number satisfying the equation `z + sqrt(2) |(z + 1)| + i` = 0.
Where does z lie, if `|(z - 5i)/(z + 5i)|` = 1.
The real value of α for which the expression `(1 - i sin alpha)/(1 + 2i sin alpha)` is purely real 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 ______.
If `(3 + i)(z + barz) - (2 + i)(z - barz) + 14i` = 0, then `barzz` is equal to ______.
If `(x + iy)^(1/5)` = a + ib, and u = `x/a - y/b`, then ______.
A complex number z is moving on `arg((z - 1)/(z + 1)) = π/2`. If the probability that `arg((z^3 -1)/(z^3 + 1)) = π/2` is `m/n`, where m, n ∈ prime, then (m + n) is equal to ______.
The smallest positive integer n for which `((1 + i)/(1 - i))^n` = –1 is ______.
Let `(-2 - 1/3i)^2 = (x + iy)/9 (i = sqrt(-1))`, where x and y are real numbers, then x – y equals 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)`
Evaluate the following:
i35
