Question

The equation \[\overline b z + \overline z b = c\]where b is a non-zero complex constant and c is real, represents
A) A circle
B) A straight line
C) A parabola
D) None of these

Answer 1

Hint: in this question we have to find locus of the point \[z\] which satisfy the given condition. First write the given complex number as a combination of real and imaginary number. Then put value of z and b in given equation.

Formula Used: Equation of complex number is given by
\[z = x + iy\]
\[b = {b_1} + i{b_2}\]
Where
z and b is a complex number
x, \[{b_1}\] represent real part of complex number
iy , \[i{b_2}\] are imaginary part of complex number
i is iota
Square of iota is equal to the negative of one

Complete step by step solution: Given: equation in form of complex number
Now we have equation \[\overline b z + \overline z b = c\]
We know that complex number is written as a combination of real and imaginary number.
\[z = x + iy\]
\[b = {b_1} + i{b_2}\]
Where
Z and b are complex number
x, \[{b_1}\] represent real part of complex number
iy , \[i{b_2}\] are imaginary part of complex number
i is iota
\[\overline z = x - iy\]
\[\overline b = {b_1} - i{b_2}\]
Put these value in\[\overline b z + \overline z b = c\]
\[({b_1} - i{b_2})(x + iy) + ({b_1} + i{b_2})(x - iy) = c\]
On simplification we get
\[2{b_1}x + 2{b_2}y = c\]
This equation represents the equation of Line.

Option ‘B’ is correct

Note: Complex number is a number which is a combination of real and imaginary number. So in complex number question we have to represent number as a combination of real and its imaginary part. Imaginary part is known as iota. Square of iota is equal to negative one.