Interpret the loci | z + i | ≥ | z + 2 | in z ∈ C . Also indicate on complex plane.

Given, | z + i | ≥ | z + 2 | Putting z = x + i y , we get ⇒ | x + i y + i | ≥ | x + i y + 2 | ⇒ | x + i y + 1 | ≥ | x + 2 + i y | ⇒ | x + i y + 1 | 2 ≥ | x + 2 + i y | 2 (Modulus is always non -negative ) ⇒ x 2 + y + 1 2 2 ≥ x + 2 2 + y 2 2 ∵ | x + i y | = x 2 + y 2 ⇒ x 2 + y + 1 2 ≥ x + 2 2 + y 2 ⇒ x 2 + y 2 + 2 y + 1 ≥ x 2 + 4 x + 4 + y 2 (Subtract x 2 and y 2 on each side of inequality) ⇒ 2 y + 1 ≥ 4 x + 4 ⇒ 0 ≥ 4 x + 4 - 2 y - 1 (Subtract 1 and 2 y on each side of inequality) ⇒ 4 x + 4 - 2 y - 1 ≤ 0 ⇒ 4 x - 2 y + 3 ≤ 0 , it represents half of the plane that lies to the left of the line 4 x - 2 y + 3 = 0 . (including the points on the line) This can ben seen in a figure.

Interpret the loci | z - 2 i | = | z + 2 i | in z ∈ C . Also indicate on complex plane.

Given, | z - 2 i | = | z + 2 i | Putting z = x + i y , we get ⇒ | x + i y - 2 i | = | x + i y + 2 i | ⇒ | x + i y - 2 | = | x + i y + 2 | ⇒ x 2 + y - 2 2 = x 2 + y + 2 2 ∵ | x + i y | = x 2 + y 2 ⇒ x 2 + y - 2 2 2 = x 2 + y + 2 2 2 (Apply squaring on both sides) ⇒ x 2 + y - 2 2 = x 2 + y + 2 2 ⇒ y - 2 2 = y + 2 2 ⇒ y 2 - 4 y + 4 = y 2 + 4 y + 4 ⇒ - 4 y = 4 y ⇒ 8 y = 0 ⇒ y = 0 , which represents equation of x - axis .

Find the locus of a complex number z = x + i y satisfying the relation z - 2 i z + 2 i = 2 . Illustrate the locus of z in the Argand plane.

Given, z - 2 i z + 2 i = 2 ⇒ z - 2 i z + 2 i = 2 ∵ z 1 z 2 = z 1 z 2 , z 2 ≠ 0 ⇒ z - 2 i = 2 z + 2 i Putting z = x + i y , we get ⇒ | x + i y - 2 i | = 2 | x + i y + 2 i | ⇒ | x + i y - 2 | = 2 | x + i y + 2 | ⇒ x 2 + y - 2 2 = 2 x 2 + y + 2 2 ∵ | x + i y | = x 2 + y 2 ⇒ x 2 + y - 2 2 2 = 2 x 2 + y + 2 2 2 (Apply squaring on both sides) ⇒ x 2 + y - 2 2 = 2 x 2 + y + 2 2 ⇒ x 2 + y 2 - 4 y + 4 = 2 x 2 + y 2 + 4 y + 4 ⇒ x 2 + y 2 - 4 y + 4 = 2 x 2 + 2 y 2 + 8 y + 8 ⇒ 0 = x 2 + y 2 + 12 y + 4 ⇒ x 2 + y 2 + 12 y + 4 = 0 ⇒ x 2 + y + 6 2 - 36 + 4 = 0 ⇒ x 2 + y + 6 2 = 32 ⇒ x 2 + y + 6 2 = 4 2 2 . It represents the circle with centre 0 , - 6 and radius 4 2 . This can be seen in a figure,

Find the locus of a complex number z = x + i y satisfying the relation | 3 z - 4 i | ≤ | 3 z + 2 | . Illustrate the locus of z in the Argand plane.

Given, | 3 z - 4 i | ≤ | 3 z + 2 | Putting z = x + i y , we get ⇒ | 3 x + i y - 4 i | ≤ | 3 x + i y + 2 | ⇒ | 3 x + 3 i y - 4 i | ≤ | 3 x + 3 i y + 2 | ⇒ | 3 x + 3 i y - 4 i | ≤ | 3 x + 3 i y + 2 | ⇒ | 3 x + i 3 y - 4 | ≤ | 3 x + 2 + i 3 y | ⇒ 3 x 2 + 3 y - 4 2 ≤ 3 x + 2 2 + 3 y 2 ∵ | x + i y | = x 2 + y 2 ⇒ 9 x 2 + 9 y 2 - 24 y + 16 ≤ 9 x 2 + 12 x + 4 + 9 y 2 Since, the square root is always positive. ⇒ 9 x 2 + 9 y 2 - 24 y + 16 ≤ 9 x 2 + + 12 x + 4 + 9 y 2 ⇒ - 24 y + 16 ≤ 12 x + 4 ⇒ - 24 y - 12 x + 16 - 4 ≤ 0 (subtracted 12 x and 4 on each side of inequality) ⇒ 24 y + 12 x - 12 ≥ 0 (Multiplied by - 1 on each side of inequality) ⇒ x + 2 y - 1 ≥ 0 (Divided by 12 on each side of inequality) It represents half of the plane that lies 'above' the line x + 2 y - 1 = 0 (including the points on the line) shown in the adjoining figure.

If z = x + i y and z 1 = 1 + 2 i , determine the region in the complex plane represented by 1 < z - z 1 ≤ 3 . Represent it on the Argand plane.

Given, z = x + i y and z 1 = 1 + 2 i And also, 1 < z - z 1 ≤ 3 ⇒ 1 < x + i y - 1 + 2 i ≤ 3 ⇒ 1 < x - 1 + i y - - 2 i ≤ 3 ⇒ 1 < x - 1 + i y - 2 ≤ 3 ⇒ 1 < x - 1 2 + y - 2 2 ≤ 3 Apply squaring on both sides of inequality, we get ⇒ 1 < x - 1 2 + y - 2 2 ≤ 9 . (Since both sides of inequality has positive real numbers) It represents circular annulus lying between concentric circles of radius 1 and 3 centred at 1 , 2 . The region includes all points on the circumference of the outer circle but excludes all points on the circumference of the inner circle. This can be seen in a figure,

E M B I B E

Understanding ISC Mathematics Class 11 Volume 1>Chapter 5 - Complex Numbers>EXERCISE 5.4

M L Aggarwal Solutions for Chapter: Complex Numbers, Exercise 4: EXERCISE 5.4

Author:M L Aggarwal

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Chapter: Complex Numbers, Exercise 4: EXERCISE 5.4

Attempt the practice questions on Chapter 5: Complex Numbers, Exercise 4: EXERCISE 5.4 with hints and solutions to strengthen your understanding. Understanding ISC Mathematics Class 11 Volume 1 solutions are prepared by Experienced Embibe Experts.

Questions from M L Aggarwal Solutions for Chapter: Complex Numbers, Exercise 4: EXERCISE 5.4 with Hints & Solutions

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 5 - Complex Numbers>EXERCISE 5.4>Q 2

Interpret the loci $| z + i | \geq | z + 2 |$ in $z \in C$ . Also indicate on complex plane.

MEDIUM

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 5 - Complex Numbers>EXERCISE 5.4>Q 2

Interpret the loci $| z - 2 i | = | z + 2 i |$ in $z \in C$ . Also indicate on complex plane.

MEDIUM

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 5 - Complex Numbers>EXERCISE 5.4>Q 3

Find the locus of a complex number $z = x + i y$ satisfying the equation $| z + i | = | z + 2 |$ . Illustrate the locus of $z$ in the Argand plane.

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 5 - Complex Numbers>EXERCISE 5.4>Q 4

Find the locus of a complex number $z = x + i y$ satisfying the relation $\Rightarrow 4 x + 4 - 2 y - 1 \leq 0$ . Illustrate the locus of $\Rightarrow 4 x - 2 y + 3 \leq 0$ in the Argand plane.

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 5 - Complex Numbers>EXERCISE 5.4>Q 5

Sketch in the complex plane the set of points $z$ satisfying $|\frac{z - 3}{z + 1}| = 3$ .

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 5 - Complex Numbers>EXERCISE 5.4>Q 6

Find the locus of a complex number $z = x + i y$ satisfying the relation $| 3 z - 4 i | \leq | 3 z + 2 |$ . Illustrate the locus of $z$ in the Argand plane.

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 5 - Complex Numbers>EXERCISE 5.4>Q 7

Find the centre and the radius of the circle represented by the equation $z \bar{z} - (2 + 3 i) z - (2 - 3 i) \bar{z} + 9 = 0$ .

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 5 - Complex Numbers>EXERCISE 5.4>Q 8

If $z = x + i y and z_{1} = 1 + 2 i$ , determine the region in the complex plane represented by $1 < |z - z_{1}| \leq 3$ . Represent it on the Argand plane.

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