#### If Re  where  then the point  lies on a : Option: 1 circle whose centre is at  Option: 2 straight line whose slope is  Option: 3 circle whose diameter is  Option: 4 straight line whose slope is

Conjugate of complex numbers and their properties -

The complex conjugate of a complex number a + ib (a, b are real numbers and b ≠ 0) is a − ib.

It is denoted as  .

i.e. if z = a + ib, then its conjugate is   = a - ib.

Conjugate of complex numbers is obtained by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged.

Note:

• When a complex number is added to its complex conjugate, the result is a real number. i.e. z = a + ib,   = a - ib

Then the sum, z + = a + ib + a - ib = 2a (which is real)

• When a complex number is multiplied by its complex conjugate, the result is a real number i.e. z = a + ib,  = a - ib

Then the product, z? = (a + ib)?(a - ib) = a2 - (ib)2

= a2 +  b2 (which is real)

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Circle(Definition) -

General Form:

The equation of a circle with centre at (h,k) and radius r is

This is known as the general equation of the circle.

Compare eq (i) and eq (ii)

h = -g, k = -h   and c=h2+k2-r2

Coordinates of the centre  (-g,-f)

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Correct Option (3)