Question

if z = x + Iy lies in third quadrant then conjugate z lies in​

Answer 1

SOLUTION

GIVEN

z = x + iy lies in third quadrant

TO DETERMINE

The conjugate z lies in

CONCEPT TO BE IMPLEMENTED

A complex number z = a + ib is defined as an ordered pair of Real numbers ( a, b) that satisfies the following conditions :

(i) Condition for equality :

(a, b) = (c, d) if and only if a = c, b = d

(ii) Definition of addition :

(a, b) + (c, d) = (a+c, b+ d)

(iii) Definition of multiplication :

(a, b). (c, d) = (ac-bd , ad+bc )

Of the ordered pair (a, b) the first component a is called Real part of z and the second component b is called Imaginary part of z

EVALUATION

We know that z = x + iy represents the point (x, y) in complex plane

Here it is given that z = x + iy lies in third quadrant

So (x, y) lies in third quadrant

∴ x and y both are negative

Now

$\sf{ \bar{z} = x - iy}$

$\sf{ \therefore \: \bar{z} \: \: represents \: the \: point \: ( x ,- y)}$

Since x and y both are negative

∴ x is negative and - y is positive

∴ The point (x, - y) second quadrant

∴ Conjugate of z lies in Second quadrant

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