Given that x , y ∈ R , solve : 4 x 2 + 3 x y + 2 x y - 3 x 2 i = 4 y 2 - x 2 2 + 3 x y - 2 y 2 i

We know that if Z 1 = Z 2 , then Re Z 1 = Re Z 2 & Im Z 1 = Im Z 2 . Given, 4 x 2 + 3 x y + 2 x y - 3 x 2 i = 4 y 2 - x 2 2 + 3 x y - 2 y 2 i ⇒ 4 x 2 + 3 x y = 4 y 2 - x 2 2 & 2 x y - 3 x 2 = 3 x y - 2 y 2 ⇒ 9 x 2 + 6 x y = 8 y 2 & 2 y 2 = x y + 3 x 2 ⇒ 9 x y 2 + 6 x y - 8 = 0 & 3 x y 2 + x y - 2 = 0 ⇒ 9 x y 2 + 12 x y - 6 x y - 8 = 0 & 3 x y 2 + 3 x y - 2 x y - 2 = 0 ⇒ 3 x y 3 x y + 4 - 2 3 x y + 4 = 0 & 3 x y x y + 1 - 2 x y + 1 = 0 ⇒ 3 x y - 2 3 x y + 4 = 0 & 3 x y - 2 x y + 1 = 0 Hence, the given expression is possible only when 3 x y - 2 = 0 . ⇒ 3 x = 2 y ⇒ x = K & y = 3 2 K , K ∈ R .

Interpret the following locii in z ∈ C . (a) 1 < | z - 2 i | < 3

Given, 1 < | z - 2 i | < 3 Put z = x + i y ⇒ 1 < | x + y - 2 i | < 3 ⇒ 1 < x 2 + y - 2 2 < 3 ⇒ 1 2 < x 2 + y - 2 2 < 3 2 Hence, z lies between two concentric circles of centre 0 , 2 and radii 1 & 3 .

Interpret the following locii in z ∈ C . (b) Re z + 2 i i z + 2 ≤ 4 , ( z ≠ 2 i )

Given, Re z + 2 i i z + 2 ≤ 4 , ( z ≠ 2 i ) Put, z = x + i y ⇒ Re x + y + 2 i 2 - y + i x ≤ 4 ⇒ Re x + y + 2 i 2 - y + i x × 2 - y - i x 2 - y - i x ≤ 4 ⇒ Re 2 x - x y + x y + 2 x + 4 - y 2 - x 2 i 2 - y 2 + x 2 ≤ 4 ⇒ 4 x 2 - y 2 + x 2 ≤ 4 ⇒ x ≤ x 2 + y - 2 2 ⇒ x - 1 2 2 + y - 2 2 ≥ 1 2 2 Hence, z lies on and outside the circle of centre 1 2 , 2 and radius 1 2 unit.

Interpret the following locii in z ∈ C . (c) Arg ( z + i ) - Arg ( z - i ) = π 2

Given, Arg ( z + i ) - Arg ( z - i ) = π 2 ⇒ Arg z + i z - i = π 2 Put, z = x + i y ⇒ Arg x + i y + 1 x + i y - 1 = π 2 ⇒ Arg x + i y + 1 x + i y - 1 × x - i y - 1 x - i y - 1 = π 2 ⇒ Arg x 2 + y 2 - 1 + i x - x y + x y + x x 2 + y - 1 2 = π 2 ⇒ Arg x 2 + y 2 - 1 x 2 + y - 1 2 + i 2 x x 2 + y - 1 2 = π 2 We know that if Arg Z = π 2 , then Re Z = 0 & Im Z > 0 ∴ x 2 + y 2 - 1 x 2 + y - 1 2 = 0 & 2 x x 2 + y - 1 2 > 0 ⇒ x 2 + y 2 - 1 = 0 & x > 0 ⇒ x 2 + y 2 = 1 & x > 0

E M B I B E

Beta Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>EXERCISE-4>Q 2

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JEE Main/Advance

IMPORTANT

Earn 100

Find the modulus, argument and the principal argument of the complex numbers.
(b) $(\tan 1 - i)^{2}$

Important Questions on Complex Numbers

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JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>EXERCISE-4>Q 3

Find the modulus, argument and the principal argument of the complex number $z = \frac{\sqrt{5 + 12 i} + \sqrt{5 - 12 i}}{\sqrt{5 + 12 i} - \sqrt{5 - 12 i}}$