Mathematics Crash Course JEE Advanced>Chapter 12 - Complex Numbers>Exercise 1>Q 40

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If $|a z + b \bar{z} + c| = α |a| |z - z_{1}|$ , $a \neq 0, a = \bar{b}, c \in R$ , where $z_{1}$ is a given non-zero complex number and $α \in R^{+},$ then the locus of $z$ maybe

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Important Questions on Complex Numbers

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Mathematics Crash Course JEE Advanced>Chapter 12 - Complex Numbers>Exercise 1>Q 41

If

|z_{1}| = |z_{2}| = |z_{3}| = 1

and

z_{1}, z_{2}, z_{3}

are represented by the vertices of an equilateral triangle then

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Mathematics Crash Course JEE Advanced>Chapter 12 - Complex Numbers>Exercise 1>Q 42

Let

s, t, r

be non-zero complex numbers and

L

be the set of solutions

z = x + i y

(x, y \in R, i = \sqrt{- 1})

of the equation

s z + t \bar{z} + r = 0

, where

\bar{z} = x - i y

. Then, which of the following statement(s) is (are) TRUE ?

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Mathematics Crash Course JEE Advanced>Chapter 12 - Complex Numbers>Exercise 1>Q 49

If

z_{1}, z_{2} \in C, z_{1}^{2} + z_{2}^{2} \in R, z_{1} (z_{1}^{2} - 3 z_{2}^{2}) = 2

and

z_{2} (3 z_{1}^{2} - z_{2}^{2}) = 11

, then the value of

z_{1}^{2} + z_{2}^{2}

is

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Mathematics Crash Course JEE Advanced>Chapter 12 - Complex Numbers>Exercise 1>Q 50

z_{1}, z_{2} \in c

and

z_{1}^{2} + z_{2}^{2} \in R

,

z_{1} (z_{1}^{2} - 3 z_{2}^{2}) = 2, z_{2} (3 z_{1}^{2} - z_{2}^{2}) = 11

. If

z_{1}^{2} + z_{2}^{2} = λ

, then determine

λ^{2}

.

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Mathematics Crash Course JEE Advanced>Chapter 12 - Complex Numbers>Exercise 1>Q 51

For all the complex numbers

z_{1}, z_{2}

satisfying

|z_{1} - 3| = 3

and

|z_{2} - 3 + 18 i| = 2

then the minimum value of

|z_{1} - z_{2}|

is equal to

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Mathematics Crash Course JEE Advanced>Chapter 12 - Complex Numbers>Exercise 1>Q 52

If

a

and

b

are positive integers such that

N = {(a + i b)}^{3} - 107 i

is a positive integer, then find the value of

\frac{N}{2}

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Mathematics Crash Course JEE Advanced>Chapter 12 - Complex Numbers>Exercise 1>Q 53

The algebraic sum of perpendicular distances from the points

1, α_{1}, α_{2}, α_{3}, α_{4}, α_{5}, α_{6}, α_{7}

, where

α_{i}

for

i = 1, 2, \dots, 7

, are complex

8^{th}

roots of unity, from the line

(3 + 2 i) z + (3 - 2 i) \bar{z} + 5 \sqrt{13} = 0

, is equal to