For complex numbers z and w, prove that...

For complex numbers `z and w`, prove that `|z|^2w- |w|^2 z = z - w`, if and only if `z = w` or `z barw = 1`.

Topper's Solved these Questions

BINOMIAL THEOREM
CENGAGE ENGLISH|Exercise All Questions|365 Videos
CONIC SECTIONS
CENGAGE ENGLISH|Exercise All Questions|1344 Videos

Similar Questions

Explore conceptually related problems

Consider the two complex numbers z and w, such that w=(z-1)/(z+2)=a+ib, " where " a,b in R " and " i=sqrt(-1). If z=CiStheta , which of the following does hold good?

If z 1 =a+ib and z 2 =c+id are complex numbers such that ∣z 1 ∣=∣z 2 ∣=1 and Re(z 1 z 2 )=0, then the pair of complex numbers w 1 =a+ic and w 2 =b+id satisfy

Let z and w be two nonzero complex numbers such that |z|=|w| and arg(z)+a r g(w)=pidot Then prove that z=- barw dot

Represent on complex plane the complex numbers w=3 + 4i and z= 6-3i together with w +z and w-z . Obtain the modulus and argument of w and z.

Let za n dw be two nonzero complex numbers such that |z|=|w|a n d arg(z)+a r g(w)=pidot Then prove that z=- bar w dot

If z and w are two non-zero complex numbers such that z=-w.

Let z=x+iy and w=u+iv be two complex numbers, such that |z|=|w|=1 and z^(2)+w^(2)=1. Then, the number of ordered pairs (z, w) is equal to (where, x, y, u, v in R and i^(2)=-1 )

Let z_1=a+i b and z_2=c+i d are two complex number such that |z_1|=r and R e(z_1z_2)=0 . If w_1=a+i c and w_2=b+i d , then |w_2|=r (b) |w_2|=r R e(w_1w_2)=0 (d) I m(w_1w_2)=0

If z_(1) = a + ib " and " z_(2) + c id are complex numbers such that |z_(1)| = |z_(2)| = 1 and Re (z_(1)bar (z)_(2)) = 0 , then the pair of complex numbers w_(1) = a + ic " and " w_(2) = b id satisfies :

Let z and omega be complex numbers such that |z|=|omega| and arg (z) dentoe the principal of z. Statement-1: If argz+ arg omega=pi , then z=-baromega Statement -2: |z|=|omega| implies arg z-arg baromega=pi , then z=-baromega

CENGAGE ENGLISH-COMPLEX NUMBERS AND QUADRATIC EQUATIONS-All Questions

If |z|<=1,|w|<=1, then show that |z- w|^2<=(|z|-|w|)^2+(argz-argw)^2
Text Solution
|
If alpha,beta are the roots of the equation x^2-2x+3=0 obtain the equa...
Text Solution
|
For complex numbers z and w, prove that |z|^2w- |w|^2 z = z - w, if...
Text Solution
|
If alpha,beta are the roots of the equation a x^2+b x+c=0, then the va...
Text Solution
|
Let z1 \and\ z2 be the roots of the equation z^2+p z+q=0, where the co...
Text Solution
|
If a in (-1,1), then roots of the quadratic equation (a-1)x^2+a x+sqrt...
Text Solution
|
The maximum value of |a r g(1/(1-z))| for |z|=1,z!=1 is given by.
Text Solution
|
If one root is square of the other root of the equation x^2+p x+q=0, t...
Text Solution
|
If z^4+1=sqrt(3)i (A) z^3 is purely real (B) z represents the vertic...
Text Solution
|
Let alpha,beta be the roots of the quadratic equation a x^2+b x+c=0 an...
Text Solution
|
If x=a+bi is a complex number such that x^2=3+4i and x^3=2+11i, where...
Text Solution
|
Let alpha,beta be the roots of x^2-x+p=0a n dgamma,delta are roots of ...
Text Solution
|
If the difference between the roots of the equation x^2+""a x""+""1...
Text Solution
|
If f(x)=x^2+2b x+2c^2 and g(x)= -x^2-2c x+b^2 are such that min f(x...
Text Solution
|
Let z be a complex number such that the imaginary part of z is nonzero...
Text Solution
|
For the equation 3x^2+p x+3=0,p >0, if one of the root is square of th...
Text Solution
|
Let z,w be complex numbers such that barz+ibarw=0 and arg zw=pi Then a...
Text Solution
|
Let f(x)=(1+b^2)x^2+2b x+1 and let m(b) be the minimum value of f(x)do...
Text Solution
|
For any two complex numbers z1 and z2, , prove that Re (z1 z2) = Re ...
Text Solution
|
If alphaa n dbeta are the roots of the equation x^2+b x+c=0,where <mat...
Text Solution
|