If alpha" and "beta are complex numbers ...

If `alpha" and "beta` are complex numbers such that `|beta|=1, " then "|(beta - alpha)/(1-bar(alpha)beta)|=`

Text Solution

Verified by Experts

Given, `|beta|=1`
`implies" "betabar(beta)=1" or "beta=(1)/(beta)" "(1)`
Now, `|(beta-alpha)/(1-bar(alpha)beta)|=|((1)/(bar(beta))-alpha)/(1-bar(alpha)beta)|=|(1)/(bar(beta))||(1-alphabar(beta))/(1-bar(alpha)beta)|=(1)/(|beta|)|bar(1-alphabar(beta))/(1-bar(alpha)beta)|`
`=|(1-bar(alpha)beta)/(1-bar(alpha)beta)|=1`

Topper's Solved these Questions

COMPLEX NUMBERS
CENGAGE|Exercise Exercise 3.1|4 Videos
COMPLEX NUMBERS
CENGAGE|Exercise Exercise 3.2|9 Videos
COMPLEX NUMBERS
CENGAGE|Exercise Comprehension|11 Videos
CIRCLES
CENGAGE|Exercise Question Bank|16 Videos
CONIC SECTIONS
CENGAGE|Exercise Solved Examples And Exercises|87 Videos

Similar Questions

Explore conceptually related problems

If alpha and beta are non-zero real number such that 2(cos beta-cos alpha)+cos alpha cos beta=1. Then which of the following is treu?

alpha beta x ^(alpha-1) e^(-beta x ^(alpha))

If alpha" and "beta are complex cube roots of unity and x=a alpha+ b beta, y= a+b, z= a beta+b alpha , then xyz is equal to

If alpha and beta are the roots of the equation x^(2) -1 =0 " then " (2alpha)/(beta ) + (2 beta)/(alpha) is equal to ..........

If alpha and beta are the roots of the equation x^(2)-ax+b=0 ,find Q (a)(alpha)/(beta)+(beta)/(alpha)

If alpha and beta are the roots of the equation x^2 + 2x +8=0 then the value of (alpha)/(beta) + (beta)/(alpha) is ………………….. .

If alpha and beta are acute such that alpha+beta and alpha-beta satisfy the equation tan^(2)theta-4tan theta+1=0 , then (alpha, beta ) =

Q. Let p and q real number such that p!= 0 , p^2!=q and p^2!=-q . if alpha and beta are non-zero complex number satisfying alpha+beta=-p and alpha^3+beta^3=q , then a quadratic equation having alpha/beta and beta/alpha as its roots is

If alpha" and "beta are the roots of the equation x^(2)+3x-4=0 , then (1)/(alpha)+(1)/(beta) is equal to

Find the centre and radius of the circle formed by all thepoints represented by z = x + iy satisfying the relation |(z-alpha)/(z-beta)|= k (k !=1) , where alpha and beta are the constant complex numbers given by alpha = alpha_1 + ialpha_2, beta = beta_1 + ibeta_2 .

CENGAGE-COMPLEX NUMBERS-Examples

If z=costheta+isintheta is a root of the equation a0z^n+a2z^(n-2)++a(n...
Text Solution
|
If |z1|=1,|z2|=2,|z3|=3,a n d|9z1z2+4z1z3+z2z3+3|=12 , then find the v...
Text Solution
|
If alpha" and "beta are complex numbers such that |beta|=1, " then "|(...
Text Solution
|
Prove that |z1+z2|^2=|z1|^2, ifz1//z2 is purely imaginary.
Text Solution
|
Let |(( z )1-2( z )2)//(2-z1( z )2)|=1a n d|z2|!=1,w h e r ez1a n d...
Text Solution
|
If z1a n dz2 are two complex numbers and c >0 , then prove that |z1+z2...
Text Solution
|
If z1, z2, z3, z4 are the affixes of four point in the Argand plane, z...
Text Solution
|
if |z1+z2|=|z1|+|z2|, then prove that a r g(z1)=a r g(z2) if |z1-z2|=...
Text Solution
|
Show that the area of the triangle on the Argand diagram formed by the...
Text Solution
|
Find the minimum value of |z-1 if ||z-3|-|z+1||=2.
Text Solution
|
Find the greatest and the least value of |z1+z2| ifz1=24+7ia n d|z2|=6...
Text Solution
|
If z is a complex number, then find the minimum value of |z|+|z-1|+|2z...
Text Solution
|
If |z1-1|lt=,|z2-2|lt=2,|z(33)|lt=3, then find the greatest value of |...
Text Solution
|
Prove that following inequalities: (i) |(z)/(|z|) -1| le |arg z| (...
Text Solution
|
Identify the locus of z if z = a +(r^2)/(z-a),> 0.
Text Solution
|
If z is any complex number such that |3z-2|+|3z+2|=4 , then identify t...
Text Solution
|
If |z|=1 and let omega=((1-z)^2)/(1-z^2) , then prove that the locus o...
Text Solution
|
Let z be a complex number having the argument theta,0<theta<pi//2,a n ...
Text Solution
|
How many solutions the system of equations ||z + 4 |-|z-3i|| =5 and...
Text Solution
|
Prove that |Z-Z1|^2+|Z-Z2|^2=a will represent a real circle [with cent...
Text Solution
|