The conjugated line of a straight line is the line that one becomes by taking the complex conjugate

Complex conjugate

In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...

 of each point on this line. One can prove that this is the same as taking the complex conjugate

Complex conjugate

In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...

s of the coefficients of this line.

So if the equation of D is D : ax + by + cz = 0, then the equation of its conjugate D* is D* : a*x + b*y + c*z = 0.

The conjugate of a real line

Real line

In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...

is the line itself.
The intersection point of two conjugated lines is always real.