If z = x + i y and arg z - 2 z + 2 = π 6 , then find the locus of z .

a r g z - 2 z + 2 = π 6 ⇒ a r g x + i y - 2 x + i y + 2 = π 6 ⇒ a r g x - 2 + i y x + 2 + i y = π 6 ⇒ a r g x - 2 + i y - a r g x + 2 + i y = π 6 ⇒ tan - 1 y x - 2 - tan - 1 y x + 2 = π 6 ⇒ tan - 1 y x - 2 - y x + 2 1 + y x - 2 . y x + 2 = π 6 ∵ tan - 1 x - tan - 1 y = tan - 1 x - y 1 + x y ; x y < - 1 ⇒ x y + 2 y - x y + 2 y x 2 - 4 + y 2 = tan π 6 ⇒ 4 y x 2 + y 2 - 4 = 1 3 ⇒ x 2 + y 2 - 4 3 y - 4 = 0 Hence, locus of z is a circle whose centre is at 0 , 2 3 & radius = 0 2 + 2 3 2 - - 4 = 4 unit

Simplify : i log x - i x + i .

i log x - i x + i = i log e x - i - i log e x + i . . . 1 As, log e z = log e r . e i θ ∵ z = r e i θ ; θ → Principle argument = log e r + log e e i θ = log e r + i θ log e e = log e z + i θ Now, ∴ log e x - i = log e x 2 + 1 + i - tan - 1 1 x = log e x 2 + 1 - i cot - 1 x ∵ tan - 1 1 x = cot - 1 x ; x > 0 log e x + i = log e x 2 + 1 + i cot - 1 x ∴ i log e x - i - i log x + 1 = i log e x 2 + 1 - i cot - 1 x - i log e x 2 + 1 + i cot - 1 x = i log e x 2 + 1 - i 2 cot - 1 x - i log e x 2 + 1 - i 2 cot - 1 x = 2 cot - 1 x , x > 0

Find the set of values of K for which the equation z z ¯ + - 3 + 4 i z ¯ - 3 + 4 i z + K = 0 represents a circle.

Given:- z z ¯ + - 3 + 4 i z ¯ - 3 + 4 i z + K = 0 represents a circle. Let z = x + i y ∴ Equation becomes x + i y x - i y + - 3 + 4 i x - i y - 3 + 4 i x + i y + K = 0 ⇒ x 2 + y 2 + - 3 x + i 3 y + 4 i x + 4 y - 3 x + i 3 y + 4 i x - 4 y + K = 0 ⇒ x 2 + y 2 - 6 x + 8 y + K = 0 . . . 1 As, we know that if x 2 + y 2 + 2 g x + 2 f y + c = 0 represents a circle then g 2 + f 2 - c ≥ 0 ∴ Given equation 1 represents a circle. ∴ - 3 2 + 4 2 - K ≥ 0 ⇒ K ≤ 25 ∴ K ∈ - ∞ , 25

Let s , t , r be non-zero complex numbers and L be the set of solutions z = x + i y x , y ∈ ℝ , i = - 1 of the equation s z + t z ¯ + r = 0 , where z ¯ = x - i y . Then, which of the following statement(s) is (are) TRUE?

If L has more than one element, then L has infinitely many elements

If s = t , then L has infinitely many elements

E M B I B E

Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>EXERCISE>Q 12

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IMPORTANT

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If $z_{1}, z_{2}, z_{3}, z_{4}$ are the affixes of four points in the Argand plane and $z$ is the affix of a point such that $|z - z_{1}| = |z - z_{2}| = |z - z_{3}| = |z - z_{4}|,$ then $z_{1}, z_{2}, z_{3}, z_{4}$ are

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Important Questions on Complex Numbers

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IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>EXERCISE>Q 13

If

A (z_{1})

and

B (z_{2})

are two fixed points in the argand plane the locus of point

P (z)

satisfying

|z - z_{1}| + |z - z_{2}| = |z_{1} - z_{2}|

, is

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>EXERCISE>Q 14

The vertices of a square are

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

and

z_{4}

taken in the anticlockwise order, then

z_{3} =

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>EXERCISE>Q 16

A rectangle is constructed in the complex plane with its sides parallel to the axes and its centre is situated at the origin. If one of the vertices of the rectangle is