# Complex Numbers - 6

since z^{2} + 4 is a quadratic expression we get a Linear remainder so

f(z) = g(z)(z ^{2} + 4 ) + az + b

We have f(2i) = i = 0 + 2ia + b ………………………………………… (1)

f(-2i) = 2i = 0 + (-2i)a + b …………………………………………… (2)

Solving equation (1) & (2) we get

Hence the remainder is az + b

i.e.

Q. 5. If z is unimodular & then prove that (n amp(z))

Ans:- If z is unimodular let z =

then

Hence

Q.6. Prove the if

Ans:- We can write

We can also

Squaring eqn. (1) we get

Take

Now write & we get

Q.1. Find all the circle which are orthogonal to |z| = 3|z - 1| = 4

Ans:-

We have to find out the circles which are orthogonal to above these two circles i.e. tangents at the point of contact are at right angle to each other we know that

r_{1}^{2} + r_{2}^{2} = (C_{1}C_{2})^{2} (Applying pythogorus Theorem in PC_{1}C_{2}

Let the circle cuts orthogonally to |z| = 3 then

Also given that circle cuts |z - 1| = 4 orthogohally . so

Now

So we can write where + is any variable which corresponds to all the circles of family

Hence the requred equalion for family curves is

&

Hence K^{2} = t^{2}

equation is

Q.2. Solve equation z^{14} - 1 = 0 & deduce taht

Ans:-

k = 0, 1, 2, ………………….. , 13 are the roots of equation.

So roots are

Hence we can write

Diuiding both sides by z we get

Now put

We know that

If the prove the points representing 4 complex number in Argand plane are concyclic .

Ans:- Let

We get

Let

Now

Squaring and adding, rR K^{2} …………………………………………… (1)

Dividing, we get,

Let equation of circle passing through

Now , satisfy the equation.

So, we have,

Adding (3) and (4),we get

s And therefore,

(By using (2))

Taking conjugate , we get

By adding (7) and (8) we get

Put these values in (5) to get

(from (6))

Thus also satisfies the equation of circle.

Hence the problem.

__Dumb Question__:- How can we assure that circle passes through

__Ans__:- Well when three pointin a plane are given a unique circle passing through those 3 point can be trawn and here

is that circle only.

1. Iota, i

2. Imaginary Part

3. Real Part

4. Complex Number

5. Argand Plane

6. Real axis

7. Imaginary axis

8. Conjugate

9. Modules

10. Argument

11. Principal Argument

12. De - movier’s Theorem

13. n^{th} root of unity

14. Rotation Theorem

15. w. cube root of unity

16. Amplitude