Complex Numbers

Algebra Level 3

If z z ˉ + ( 3 − i ) z + ( 3 + i ) z ˉ + 1 = 0 z\bar { z } +(3-i)z+(3+i)\bar { z } +1=0 represents a circle , then the coordinates of its center and its radius respectively are

( − 3 , 1 ) , 4 (-3,1) , 4 ( − 3 , − 1 ) , 4 (-3,-1) , 4 ( − 3 , − 1 ) , 3 (-3,-1) , 3 ( − 3 , 1 ) , 3 (-3,1) , 3

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Sahil Bansal by Sahil Bansal , 21, India

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1 solution

Let z = x + iy ⇒ \Rightarrow z z ˉ + ( 3 − i ) z + ( 3 + i ) z ˉ + 1 z\bar{z} + (3-i)z + (3+i)\bar{z} + 1 = 0 = x 2 + y 2 + 6 x + 2 y + 1 = ( x + 3 ) 2 + ( y + 1 ) 2 − 9 x^2 + y^2 + 6x + 2y + 1 = (x + 3)^{2} + (y + 1)^2 - 9 ⇒ \Rightarrow ( x + 3 ) 2 + ( y + 1 ) 2 = 9 (x + 3)^{2} + (y + 1)^2 = 9 ⇒ \Rightarrow This is an equation of a circle of radius 3 and centre (-3,-1) .

Note.- If a, b complex numbers then a b + a ˉ b ˉ = 2 R e ( a b ) ab + \bar{a}\bar{b} = 2Re(ab)

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